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If the definition of a large cardinal notion $\phi(\kappa)$ asserts the existence of a cardinal $\theta \gt \kappa$, one can define a large cardinal notion asserting that there are unboundedly many such $\theta$, that is, for every ordinal $\gamma$ there is such a $\theta$ (is that what you mean by superfication?):

• If there is a $\theta$ such that $V_\kappa$ is an elementary submodel of $V_\theta$, $\kappa$ is said to be 0-extendible or otherwordly; if additionally $\kappa$ is inaccessible, it is said to be 0-pseudo-uplifting; and if additionally $\kappa$ and $\theta$ are both inaccessible, it is said to be 0-uplifting. If there are unboundely many such $\theta$, $\kappa$ is said to be totally otherwordly, pseudo-uplifting or uplifting, respectively.
• If for every $A \subseteq V_\kappa$, there exists transitive models $M$ and $N$ and an elementary embedding $j: M \to N$ such that the critical point of $j$ is $\kappa$, $V_\kappa \subset M$, and $V_{j(\kappa)} \subset N$, then $\kappa$ is said to be weakly superstrong. If for every $\gamma$ and every $A \subseteq V_\kappa$ there exists such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be superstrongly unfoldable.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If additionally $M^\theta \subset M$, $\kappa$ is said to be high jump. If for every $\gamma$ there is a high jump embedding $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be super-high-jump. One can similarly define super-high-jump for strongness cardinals but this property is equivalent to being globally superstrong just like high jump for strongness is equivalent to superstrong.
• The definitions of superstrong, almost huge and huge cardinals involve elementary embeddings $j: V \to M$ with critical point $\kappa$ and certain other properties. If for every $\gamma$ there is such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be globally superstrong, super-almost-huge, or superhuge, respectively. One can define a similar strengthening of the definition of $\gamma$-extendible cardinals (there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$); this strengthening doesn't appear to have a name but one could call such cardinals globally $\gamma$-extendible.
• An elementary embedding $j: V_\lambda \to V_\lambda$ is called a rank into rank, $I_3$ or $E_0$ embedding. Such an elementary embedding extends to a $\Sigma_0$-elementary embedding $\hat{j}: V_{\lambda+1} \to V_{\lambda+1}$; if $\hat{j}$ is $\Sigma_{2n}$-elementary (equivalently $\Sigma_{2n-1}$-elementary) it is called an $E_n$ embedding (for $n \lt \omega$). An elementary embedding $j: V \to M$ such that $j^n(\kappa) \subset M$ for all $n \lt \omega$ is called an $I_2$ embedding (any $I_2$ embedding restricts to an $E_1$ embedding and conversely any $E_1$ embedding extends to an $I_2$ embedding) and the critical point of an $I_2$ embedding is sometimes said to be $\omega$-fold superstrong. A further strengthening of $E_n$, involving a fully elementary embedding $j: V_{\lambda+1} \to V_{\lambda+1}$, is called $I_1$ or $E_\omega$. If for every $\gamma$ there is an $E_n$ embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be a P-$E_n$ cardinal; P-$E_0$ cardinals are also called $\omega$-fold extendible. If for every $\gamma$ there is an $I_2$ ($\omega$-fold superstrong) embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be $\omega$-fold strong (which is of course equivalent to P-$E_1$). Similiarly we can define P-$E_\omega$ cardinals.

• If there is a $\theta$ such that $V_\kappa$ is an elementary submodel of $V_\theta$, $\kappa$ is said to be otherwordly. One can define $C^{(n)}$-otherwordly cardinals by additionally requiring that $\theta$ is $\Sigma_n$-correct.
• The definitions of superstrong, $2$-fold $\gamma$-strong, almost huge, huge and $\lambda$-hyperhuge cardinals involve elementary embeddings $j: V \to M$ with critical point $\kappa$ and the definition of $\gamma$-extendible cardinals says that there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$. If additionally $\theta$ is $\Sigma_n$-correct, $\kappa$ is said to be $C^{(n)}$-superstrong, $C^{(n)}$-almost huge, $C^{(n)}$-huge, of $C^{(n)}$-$\gamma$-extendible, respectively, and one can similarly define $C^{(n)}$-$2$-fold $\gamma$-strong and $C^{(n)}$-$\lambda$-hyperhuge cardinals.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If additionally $M^\theta \subset M$, $\kappa$ is said to be high jump. One can define $C^{(n)}$-high jump for strongness and $C^{(n)}$-high jump cardinals by additionally requiring that $\theta$ is $\Sigma_n$-correct. Just like high jump for strongness is equivalent to superstrong, $C^{(n)}$-high jump for strongness is equivalent to $C^{(n)}$-superstrong.
• If for every $A \subseteq V_\kappa$, there exists transitive models $M$ and $N$ and an elementary embedding $j: M \to N$ such that the critical point of $j$ is $\kappa$, $V_\kappa \subset M$, and $V_{j(\kappa)} \subset N$, then $\kappa$ is said to be weakly superstrong. One can define $C^{(n)}$-weakly superstrong cardinals by requiring that for every $A$ there is such a $j$ with $j(\kappa)$ $\Sigma_n$-correct.
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$, then $\kappa$ is said to be a Shelah cardinal. If $M^{j(f)(\kappa)} \subset M$, $\kappa$ is said to be Shelah for supercompactness. One can define $C^{(n)}$-Shelah cardinals as follows: for every function $f: \kappa \to \kappa$ such that all ordinals in the range of $f$ are $\Sigma_n$-correct in $V_\kappa$, there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$ and $j(f)(\kappa)$ is $\Sigma_n$-correct. One can similarly define $C^{(n)}$-Shelah for supercompactness cardinals.
• If there is an $E_i$ embedding $j: V_\lambda \to V_\lambda$ with critical point $\kappa$ such that $j^m(\kappa)$ is $\Sigma_n$-correct, then $\kappa$ is said to be $m$-$C^{(n)}$-$E_i$; if $\lambda$ is $\Sigma_n$-correct, $\kappa$ is said to be $\omega$-$C^{(n)}$-$E_i$. 1-$C^{(n)}$-$E_i$ cardinals are also called $C^{(n)}$-$E_i$, 1-$C^{(n)}$-$E_0$ cardinals are also called $C^{(n)}$-$I_3$, and $\omega$-$C^{(n)}$-$E_i$ cardinals are also called $C^{(n)+}$-$I_3$. $m$-$C^{(n)}$-$I_1$ cardinals (which could also be called $m$-$C^{(n)}$-$E_\omega$) are defined similarly and 1-$C^{(n)}$-$I_1$ and $\omega$-$C^{(n)}$-$I_1$ are also called $C^{(n)}$-$I_1$ and $C^{(n)+}$-$I_1$, respectively.
• One can similarly define $m$-$C^{(n)}$ variants of $k$-fold variants (see below): $\kappa$ is $m$-$C^{(n)}$-$k$-superstrong ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\gamma$-extendible ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\gamma$-strong ($1 \le m \le k-1$), $m$-$C^{(n)}$-$k$-fold high jump for extendibility ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold high jump for strongness ($1 \le m \le k-1$), $m$-$C^{(n)}$-almost-$k$-huge ($1 \le m \le k$), $m^+$-$C^{(n)}$-$k$-fold 0-extendible ($0 \le m \le k-1$; see below for $n=k$), $m$-$C^{(n)}$-$k$-huge ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\gamma$-ultrahuge ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\lambda$-hyperhuge ($1 \le m \le k$) or $m$-$C^{(n)}$-$k$-fold high jump ($1 \le m \le k-1$) if there is an elementary embedding $j$ witnessing that $\kappa$ is $k$-superstrong, $k$-fold $\gamma$-extendible, $k$-fold $\gamma$-strong, $k$-fold high jump for extendibility, $k$-fold high jump for strongness, almost-$k$-huge, $k$-fold 0-extendible, $k$-huge, $k$-fold $\gamma$-ultrahuge, $k$-fold $\lambda$-hyperhuge, or $k$-fold high jump, respectively, and additionally $j^m(\kappa)$ is $\Sigma_n$-correct.
• For $k$-fold high jump one can additionally define the following $m^+$-$C^{(n)}$ variants: $\kappa$ is $m^+$-$C^{(n)}$-$k$-fold high jump ($0 \le m \le k-1$), $m^+$-$C^{(n)}$-$k$-fold high jump for strongness ($0 \le m \le k-1$), or $m^+$-$C^{(n)}$-$k$-fold high jump for extendibility ($0 \le m \le k-1$; see below for $n=k$) if there is an elementary embedding $j$ witnessing that $\kappa$ is $k$-fold high jump, $k$-fold high jump for strongness, or $k$-fold high jump for extendibility, respectively, such that $j^m(\theta)$ is $\Sigma_n$-correct, where $\theta$ is the clearance of $j$.
• One could similarly define $m$-$C^{(n)}$-$k$-fold Shelah ($1 \le m \le k-1$) and $m^+$-$C^{(n)}$-$k$-fold Shelah ($0 \le m \le k-1$) cardinals, but for a given $k$ they all turn out to be equivalent, so such cardinals should simply be called $C^{(n)}$-$k$-fold Shelah. Similarly, there are many possible definitions of $C^{(n)}$-$k$-fold Shelah for supercompactness and $C^{(n)}$-$k$-fold Shelah for extendibility cardinals, all of which are equivalent to being $C^{(n)}$-$k$+1-fold Shelah.
• One can define that $\kappa$ is $k$-fold high jump for extendibility if it is the critical point of and elementary embedding $j: V_{j^k(\theta)} \to V_\eta$, where $\theta$ is the clearance of $j$. One can define $k^+$-$C^{(n)}$-$k$-fold high jump for extendibility cardinals by additionally requiring that $\eta$ (and thus the supremum of the ordinals in the range of $j$, if that isn't $\eta$) is $\Sigma_n$-correct. Similarly, one can define that $\kappa$ is $k^+$-$C^{(n)}$-$k$-fold 0-extendible if it is the critical point of an elementary embedding $j: V_{j^{n-1}(\kappa)} \to V_\eta$ such that $\eta$ (and thus the supremum of the ordinals in the range of $j$, if that isn't $\eta$) is $\Sigma_n$-correct.
• This method can be combined with those described above to define $C^{(n)}$-totally otherwordly, $C^{(n)}$-pseudo-uplifting, $C^{(n)}$-uplifting, $C^{(n)}$-superstrongly unfoldable, $C^{(n)}$-globally superstrong, $C^{(n)}$-extendible, $C^{(n)}$-super-high-jump, $C^{(n)}$-super-almost-huge, $C^{(n)}$-superhuge, $C^{(n)}$-ultrahuge and $C^{(n)}$-hyperhuge cardinals.

• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma} \subset M$, $\kappa$ is said to be $\gamma$-strong. If $V_{j^{n-1}(\kappa+\gamma)} \subset M$, $\kappa$ is said to be $n$-fold $\gamma$-strong.
• If $\kappa$ is $\gamma$-strong for all $\gamma$, it is said to be strong. If $\kappa$ is $n$-fold $\gamma$-strong for all $\gamma$, it is said to be $n$-fold strong.
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ whose critical point, which we will call $\mu$, is less than $\kappa$ and such that $f"\mu \subset \mu$, $V_{j(f)(\mu)} \subset M$, and $j(f)(\mu)=f(\mu)$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be a Woodin cardinal. If for every such $f$ there are such $j: V \to M$ and $\mu$ such that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent), then $\kappa$ is said to be $n$-fold Woodin.
• If for every function $f: \kappa \to \kappa$ there is an measure $U_f \subset \mathcal{P}(\kappa)$ such that for $U_f$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{j(f)(\mu)} \subset M$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be weakly hyper-Woodin. One can define $n$-fold weakly hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent).
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$, then $\kappa$ is said to be a Shelah cardinal. If for every such $f$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(j^{n-1}(\kappa))} \subset M$, then $\kappa$ is said to be a $n$-fold Shelah cardinal.
• If there is an measure $U \subset \mathcal{P}(\kappa)$ such that for every function $f: \kappa \to \kappa$, $U$ satisfies that for $U$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{f(\mu)} \subset M$ and $j(f)(\mu)=f(\mu)$, then $\kappa$ is said to be hyper-Woodin (contrary to what I previously said, we can't drop the requirement that $j(f)(\mu)=f(\mu)$, even if we replace $V_{f(\mu)} \subset M$ with $V_{j(f)(\mu)} \subset M$). One can define $n$-fold hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{f(j^{n-1}(\mu))} \subset M$ and for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. One can define $n$-fold high jump for strongness cardinals by requiring $V_{j^{n-1}(\theta)} \subset M$. For $n \ge 2$, unlike for $n=1$, $n$-fold high jump for strongness is not equivalent to $n$-fold superstrong but weaker than $n-1$-fold almost huge and stronger than $n$-fold super-high-jump.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$, then $\kappa$ is said to be superstrong. If $V_{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-superstrong.
• One can define $n$-fold globally superstrong as follows: for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$.
• If there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $\gamma$-extendible. If there is an elementary embedding $j: V_{j(\kappa+\gamma)} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $n$-fold $\gamma$-extendible.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^\lambda \subset M$, $\kappa$ is said to be $\lambda$-supercompact. If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^{n-1}(\lambda)} \subset M$, $\kappa$ is said to be $n$-fold $\lambda$-supercompact.
• If $\kappa$ is $\lambda$-supercompact for all $\gamma$, it is said to be supercompact. If $\kappa$ is $n$-fold $\lambda$-supercompact for all $\gamma$, it is said to be $n$-fold supercompact.
• If $\kappa$ is $\gamma$-extendible for all $\gamma$, it is said to be extendible. If $\kappa$ is $n$-fold $\gamma$-extendible for all $\gamma$, it is said to be $n$-fold extendible. Being $n$-fold extendible is equivalent to being $n+1$-fold strong.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $M^\theta \subset M$, $\kappa$ is said to be high jump. One can define $n$-fold high jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$, and clearance $\theta$ such that $M^\theta \subset M$ and $\theta \gt \gamma$, then $\kappa$ is said to be super-high-jump. One can define $n$-fold super-high-jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$ (and still $\theta \gt \gamma$).
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$, then $\kappa$ is said to be almost huge. If $M^{\lt j^n(\kappa)} \subset M$, $\kappa$ is said to be almost $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-huge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-$n$-huge
• An elementary embedding $j: V_{j(\kappa)} \to V_\eta$ with critical point $\kappa$ is called an $A_2$ embedding. The critical point can also be called a 2-fold 0-extendible cardinal. More generally, if $\kappa$ is the critical point of an elementary embedding $j: V_{j^{n-1}(\kappa)} \to V_\eta$, one can call it an $n$-fold 0-extendible cardinal.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$, then $\kappa$ is said to be huge. If $M^{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be superhuge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-$n$-huge
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $V_{j({\kappa+\gamma})} \subset M$, $\kappa$ is said to be $\gamma$-ultrahuge. One can define $n$-fold $\gamma$-ultrahuge by $M^{j^n(\kappa)} \subset M$ and $V_{\kappa+\gamma} \subset M$.
• If $\kappa$ is $\gamma$-ultrahuge for all $\gamma$, it is said to be ultrahuge. Thus one can define $n$-fold $\gamma$-ultrahuge to mean $n$-fold ultrahuge for all $\gamma$.
• 2-fold $\lambda$-supercompact cardinals are also called $\lambda$-hyperhuge and 2-fold supercompact cardinals are also called hyperhuge. Thus one can alternatively refer to $n+1$-fold $\lambda$-supercompact as $n$-fold $\lambda$-hyperhuge and $n+1$-fold supercompact cardinals as $n$-fold hyperhuge. For $n \ge 1$, a cardinal is $n$-fold hyperhuge iff it is $n+1$-fold extendible (thus iff it is $n+2$-fold strong).

If the definition of a large cardinal notion $\phi(\kappa)$ asserts the existence of a cardinal $\theta \gt \kappa$, one can define a large cardinal notion asserting that there are unboundedly many such $\theta$, that is, for every ordinal $\gamma$ there is such a $\theta \gt \gamma$ (is that what you mean by superfication?):

• If there is a $\theta$ such that $V_\kappa$ is an elementary submodel of $V_\theta$, $\kappa$ is said to be 0-extendible or otherwordly; if additionally $\kappa$ is inaccessible, it is said to be 0-pseudo-uplifting; and if additionally $\kappa$ and $\theta$ are both inaccessible, it is said to be 0-uplifting. If there are unboundely many such $\theta$, $\kappa$ is said to be totally otherwordly, pseudo-uplifting or uplifting, respectively.
• If for every $A \subseteq V_\kappa$, there exists transitive models $M$ and $N$ and an elementary embedding $j: M \to N$ such that the critical point of $j$ is $\kappa$, $V_\kappa \cup \{A\} \subset M$, and $V_{j(\kappa)} \subset N$, then $\kappa$ is said to be weakly superstrong. If for every $\gamma$ and every $A \subseteq V_\kappa$ there exists such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be superstrongly unfoldable.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If additionally $M^\theta \subset M$, $\kappa$ is said to be high jump. If for every $\gamma$ there is a high jump embedding $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be super-high-jump. One can similarly define super-high-jump for strongness cardinals but this property is equivalent to being globally superstrong just like high jump for strongness is equivalent to superstrong.
• The definitions of superstrong, almost huge and huge cardinals involve elementary embeddings $j: V \to M$ with critical point $\kappa$ and certain other properties. If for every $\gamma$ there is such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be globally superstrong, super-almost-huge, or superhuge, respectively. One can define a similar strengthening of the definition of $\gamma$-extendible cardinals (there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$); this strengthening doesn't appear to have a name but one could call such cardinals globally $\gamma$-extendible.
• An elementary embedding $j: V_\lambda \to V_\lambda$ is called a rank into rank, $I_3$ or $E_0$ embedding. Such an elementary embedding extends to a $\Sigma_0$-elementary embedding $\hat{j}: V_{\lambda+1} \to V_{\lambda+1}$; if $\hat{j}$ is $\Sigma_{2n}$-elementary (equivalently $\Sigma_{2n-1}$-elementary) it is called an $E_n$ embedding (for $n \lt \omega$). An elementary embedding $j: V \to M$ such that $j^n(\kappa) \subset M$ for all $n \lt \omega$ is called an $I_2$ embedding (any $I_2$ embedding restricts to an $E_1$ embedding and conversely any $E_1$ embedding extends to an $I_2$ embedding) and the critical point of an $I_2$ embedding is sometimes said to be $\omega$-fold superstrong. A further strengthening of $E_n$, involving a fully elementary embedding $j: V_{\lambda+1} \to V_{\lambda+1}$, is called $I_1$ or $E_\omega$. If for every $\gamma$ there is an $E_n$ embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be a P-$E_n$ cardinal; P-$E_0$ cardinals are also called $\omega$-fold extendible. If for every $\gamma$ there is an $I_2$ ($\omega$-fold superstrong) embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be $\omega$-fold strong (which is of course equivalent to P-$E_1$). Similiarly we can define P-$E_\omega$ cardinals.

• If there is a $\theta$ such that $V_\kappa$ is an elementary submodel of $V_\theta$, $\kappa$ is said to be otherwordly. One can define $C^{(n)}$-otherwordly cardinals by additionally requiring that $\theta$ is $\Sigma_n$-correct.
• The definitions of superstrong, $2$-fold $\gamma$-strong, almost huge, huge and $\lambda$-hyperhuge cardinals involve elementary embeddings $j: V \to M$ with critical point $\kappa$ and the definition of $\gamma$-extendible cardinals says that there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$. If additionally $\theta$ is $\Sigma_n$-correct, $\kappa$ is said to be $C^{(n)}$-superstrong, $C^{(n)}$-almost huge, $C^{(n)}$-huge, of $C^{(n)}$-$\gamma$-extendible, respectively, and one can similarly define $C^{(n)}$-$2$-fold $\gamma$-strong and $C^{(n)}$-$\lambda$-hyperhuge cardinals.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If additionally $M^\theta \subset M$, $\kappa$ is said to be high jump. One can define $C^{(n)}$-high jump for strongness and $C^{(n)}$-high jump cardinals by additionally requiring that $\theta$ is $\Sigma_n$-correct. Just like high jump for strongness is equivalent to superstrong, $C^{(n)}$-high jump for strongness is equivalent to $C^{(n)}$-superstrong.
• If for every $A \subseteq V_\kappa$, there exists transitive models $M$ and $N$ and an elementary embedding $j: M \to N$ such that the critical point of $j$ is $\kappa$, $V_\kappa \cup \{A\} \subset M$, and $V_{j(\kappa)} \subset N$, then $\kappa$ is said to be weakly superstrong. One can define $C^{(n)}$-weakly superstrong cardinals by requiring that for every $A$ there is such a $j$ with $j(\kappa)$ $\Sigma_n$-correct.
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$, then $\kappa$ is said to be a Shelah cardinal. If $M^{j(f)(\kappa)} \subset M$, $\kappa$ is said to be Shelah for supercompactness. One can define $C^{(n)}$-Shelah cardinals as follows: for every function $f: \kappa \to \kappa$ such that all ordinals in the range of $f$ are $\Sigma_n$-correct in $V_\kappa$, there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$ and $j(f)(\kappa)$ is $\Sigma_n$-correct. One can similarly define $C^{(n)}$-Shelah for supercompactness cardinals.
• If there is an $E_i$ embedding $j: V_\lambda \to V_\lambda$ with critical point $\kappa$ such that $j^m(\kappa)$ is $\Sigma_n$-correct, then $\kappa$ is said to be $m$-$C^{(n)}$-$E_i$; if $\lambda$ is $\Sigma_n$-correct, $\kappa$ is said to be $\omega$-$C^{(n)}$-$E_i$. 1-$C^{(n)}$-$E_i$ cardinals are also called $C^{(n)}$-$E_i$, 1-$C^{(n)}$-$E_0$ cardinals are also called $C^{(n)}$-$I_3$, and $\omega$-$C^{(n)}$-$E_i$ cardinals are also called $C^{(n)+}$-$I_3$. $m$-$C^{(n)}$-$I_1$ cardinals (which could also be called $m$-$C^{(n)}$-$E_\omega$) are defined similarly and 1-$C^{(n)}$-$I_1$ and $\omega$-$C^{(n)}$-$I_1$ are also called $C^{(n)}$-$I_1$ and $C^{(n)+}$-$I_1$, respectively.
• One can similarly define $m$-$C^{(n)}$ variants of $k$-fold variants (see below): $\kappa$ is $m$-$C^{(n)}$-$k$-superstrong ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\gamma$-extendible ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\gamma$-strong ($1 \le m \le k-1$), $m$-$C^{(n)}$-$k$-fold high jump for extendibility ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold high jump for strongness ($1 \le m \le k-1$), $m$-$C^{(n)}$-almost-$k$-huge ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold 0-extendible ($0 \le m \le k-1$; see below for $n=k$), $m$-$C^{(n)}$-$k$-huge ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\gamma$-ultrahuge ($1 \le m \le k$), $m$-$C^{(n)}$-$k$-fold $\lambda$-hyperhuge ($1 \le m \le k$) or $m$-$C^{(n)}$-$k$-fold high jump ($1 \le m \le k-1$) if there is an elementary embedding $j$ witnessing that $\kappa$ is $k$-superstrong, $k$-fold $\gamma$-extendible, $k$-fold $\gamma$-strong, $k$-fold high jump for extendibility, $k$-fold high jump for strongness, almost-$k$-huge, $k$-fold 0-extendible, $k$-huge, $k$-fold $\gamma$-ultrahuge, $k$-fold $\lambda$-hyperhuge, or $k$-fold high jump, respectively, and additionally $j^m(\kappa)$ is $\Sigma_n$-correct.
• For $k$-fold high jump one can additionally define the following $m^+$-$C^{(n)}$ variants: $\kappa$ is $m^+$-$C^{(n)}$-$k$-fold high jump ($0 \le m \le k-1$), $m^+$-$C^{(n)}$-$k$-fold high jump for strongness ($0 \le m \le k-1$), or $m^+$-$C^{(n)}$-$k$-fold high jump for extendibility ($0 \le m \le k-1$; see below for $n=k$) if there is an elementary embedding $j$ witnessing that $\kappa$ is $k$-fold high jump, $k$-fold high jump for strongness, or $k$-fold high jump for extendibility, respectively, such that $j^m(\theta)$ is $\Sigma_n$-correct, where $\theta$ is the clearance of $j$.
• One could similarly define $m$-$C^{(n)}$-$k$-fold Shelah ($1 \le m \le k-1$) and $m^+$-$C^{(n)}$-$k$-fold Shelah ($0 \le m \le k-1$) cardinals, but for a given $k$ they all turn out to be equivalent, so such cardinals should simply be called $C^{(n)}$-$k$-fold Shelah. Similarly, there are many possible definitions of $C^{(n)}$-$k$-fold Shelah for supercompactness and $C^{(n)}$-$k$-fold Shelah for extendibility cardinals, all of which are equivalent to being $C^{(n)}$-$k$+1-fold Shelah.
• One can define that $\kappa$ is $k$-fold high jump for extendibility if it is the critical point of and elementary embedding $j: V_{j^k(\theta)} \to V_\eta$, where $\theta$ is the clearance of $j$. One can define $k^+$-$C^{(n)}$-$k$-fold high jump for extendibility cardinals by additionally requiring that $\eta$ (and thus the supremum of the ordinals in the range of $j$, if that isn't $\eta$) is $\Sigma_n$-correct. Similarly, one can define that $\kappa$ is $k^+$-$C^{(n)}$-$k$-fold 0-extendible if it is the critical point of an elementary embedding $j: V_{j^{n-1}(\kappa)} \to V_\eta$ such that $\eta$ (and thus the supremum of the ordinals in the range of $j$, if that isn't $\eta$) is $\Sigma_n$-correct.
• This method can be combined with those described above to define $C^{(n)}$-totally otherwordly, $C^{(n)}$-pseudo-uplifting, $C^{(n)}$-uplifting, $C^{(n)}$-superstrongly unfoldable, $C^{(n)}$-globally superstrong, $C^{(n)}$-extendible, $C^{(n)}$-super-high-jump, $C^{(n)}$-super-almost-huge, $C^{(n)}$-superhuge, $C^{(n)}$-ultrahuge and $C^{(n)}$-hyperhuge cardinals.

• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma} \subset M$, $\kappa$ is said to be $\gamma$-strong. If $V_{j^{n-1}(\kappa+\gamma)} \subset M$, $\kappa$ is said to be $n$-fold $\gamma$-strong.
• If $\kappa$ is $\gamma$-strong for all $\gamma$, it is said to be strong. If $\kappa$ is $n$-fold $\gamma$-strong for all $\gamma$, it is said to be $n$-fold strong.
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ whose critical point, which we will call $\mu$, is less than $\kappa$ and such that $f"\mu \subset \mu$, $V_{j(f)(\mu)} \subset M$, and $j(f)(\mu)=f(\mu)$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be a Woodin cardinal. If for every such $f$ there are such $j: V \to M$ and $\mu$ such that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent), then $\kappa$ is said to be $n$-fold Woodin.
• If for every function $f: \kappa \to \kappa$ there is an measure $U_f \subset \mathcal{P}(\kappa)$ such that for $U_f$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{j(f)(\mu)} \subset M$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be weakly hyper-Woodin. One can define $n$-fold weakly hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent).
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$, then $\kappa$ is said to be a Shelah cardinal. If for every such $f$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(j^{n-1}(\kappa))} \subset M$, then $\kappa$ is said to be a $n$-fold Shelah cardinal.
• If there is an measure $U \subset \mathcal{P}(\kappa)$ such that for every function $f: \kappa \to \kappa$, $U$ satisfies that for $U$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{f(\mu)} \subset M$ and $j(f)(\mu)=f(\mu)$, then $\kappa$ is said to be hyper-Woodin (contrary to what I previously said, we can't drop the requirement that $j(f)(\mu)=f(\mu)$, even if we replace $V_{f(\mu)} \subset M$ with $V_{j(f)(\mu)} \subset M$). One can define $n$-fold hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{f(j^{n-1}(\mu))} \subset M$ and for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If there are unboundedly many cardinals that are clearances of high jump for strongness embeddings with critical point $\kappa$, $\kappa$ can be said to be super high jump for strongness One can define $n$-fold high jump for strongness cardinals by requiring $V_{j^{n-1}(\theta)} \subset M$, and one can similarly define $n$-fold super high jump for strongness cardinals. For $n \ge 2$, unlike for $n=1$, $n$-fold high jump for strongness and $n$-fold super high jump for strongness are not equivalent to $n$-fold superstrong and $n$-fold globally superstrong but weaker than $n-1$-fold almost huge and stronger than $n$-fold super-high-jump.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$, then $\kappa$ is said to be superstrong. If $V_{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-superstrong.
• One can define $n$-fold globally superstrong as follows: for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$.
• If there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $\gamma$-extendible. If there is an elementary embedding $j: V_{j(\kappa+\gamma)} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $n$-fold $\gamma$-extendible.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^\lambda \subset M$, $\kappa$ is said to be $\lambda$-supercompact. If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^{n-1}(\lambda)} \subset M$, $\kappa$ is said to be $n$-fold $\lambda$-supercompact.
• If $\kappa$ is $\lambda$-supercompact for all $\gamma$, it is said to be supercompact. If $\kappa$ is $n$-fold $\lambda$-supercompact for all $\gamma$, it is said to be $n$-fold supercompact.
• If $\kappa$ is $\gamma$-extendible for all $\gamma$, it is said to be extendible. If $\kappa$ is $n$-fold $\gamma$-extendible for all $\gamma$, it is said to be $n$-fold extendible. Being $n$-fold extendible is equivalent to being $n+1$-fold strong.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $M^\theta \subset M$, $\kappa$ is said to be high jump. One can define $n$-fold high jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$, and clearance $\theta$ such that $M^\theta \subset M$ and $\theta \gt \gamma$, then $\kappa$ is said to be super-high-jump. One can define $n$-fold super-high-jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$ (and still $\theta \gt \gamma$).
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$, then $\kappa$ is said to be almost huge. If $M^{\lt j^n(\kappa)} \subset M$, $\kappa$ is said to be almost $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-huge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-$n$-huge
• An elementary embedding $j: V_{j(\kappa)} \to V_\eta$ with critical point $\kappa$ is called an $A_2$ embedding. The critical point can also be called a 2-fold 0-extendible cardinal. More generally, if $\kappa$ is the critical point of an elementary embedding $j: V_{j^{n-1}(\kappa)} \to V_\eta$, one can call it an $n$-fold 0-extendible cardinal.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$, then $\kappa$ is said to be huge. If $M^{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be superhuge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-$n$-huge
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $V_{j({\kappa+\gamma})} \subset M$, $\kappa$ is said to be $\gamma$-ultrahuge. One can define $n$-fold $\gamma$-ultrahuge by $M^{j^n(\kappa)} \subset M$ and $V_{\kappa+\gamma} \subset M$.
• If $\kappa$ is $\gamma$-ultrahuge for all $\gamma$, it is said to be ultrahuge. Thus one can define $n$-fold $\gamma$-ultrahuge to mean $n$-fold ultrahuge for all $\gamma$.
• 2-fold $\lambda$-supercompact cardinals are also called $\lambda$-hyperhuge and 2-fold supercompact cardinals are also called hyperhuge. Thus one can alternatively refer to $n+1$-fold $\lambda$-supercompact as $n$-fold $\lambda$-hyperhuge and $n+1$-fold supercompact cardinals as $n$-fold hyperhuge. For $n \ge 1$, a cardinal is $n$-fold hyperhuge iff it is $n+1$-fold extendible (thus iff it is $n+2$-fold strong).

• If there is a $\theta$ such that $V_\kappa$ is an elementary submodel of $V_\theta$, $\kappa$ is said to be 0-extendible or otherwordly; if additionally $\kappa$ is inaccessible, it is said to be 0-pseudo-uplifting; and if additionally $\kappa$ and $\theta$ are both inaccessible, it is said to be 0-uplifting. If there are unboundely many such $\theta$, $\kappa$ is said to be totally otherwordly, pseudo-uplifting or uplifting, respectively.
• If for every $A \subseteq V_\kappa$, there exists transitive models $M$ and $N$ and an elementary embedding $j: M \to N$ such that the critical point of $j$ is $\kappa$, $V_\kappa \subset M$, and $V_{j(\kappa)} \subset N$, then $\kappa$ is said to be weakly superstrong. If for every $\gamma$ and every $A \subseteq V_\kappa$ there exists such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be superstrongly unfoldable.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If additionally $M^\theta \subset M$, $\kappa$ is said to be high jump. If for every $\gamma$ there is a high jump embedding $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be super-high-jump. One can similarly define super-high-jump for strongness cardinals but this property is equivalent to being globally superstrong just like high jump for strongness is equivalent to superstrong.
• The definitions of superstrong, almost huge and huge cardinals involve elementary embeddings $j: V \to M$ with critical point $\kappa$ and certain other properties. If for every $\gamma$ there is such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be globally superstrong, super-almost-huge, or superhuge, respectively. One can define a similar strengthening of the definition of $\gamma$-extendible cardinals (there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$); this strengthening doesn't appear to have a name but one could call such cardinals globally $\gamma$-extendible.
• An elementary embedding $j: V_\lambda \to V_\lambda$ is called a rank into rank, $I_3$ or $E_0$ embedding. A rank into rank embedding satisfying certain additional conditions is called an $E_n$ embedding (for $n \lt \omega$). An elementary embedding $j: V \to M$ such that $j^n(\kappa) \subset M$ for all $n \lt \omega$ is called an $I_2$ embedding (any $I_2$ embedding restricts to an $E_1$ embedding and conversely any $E_1$ embedding extends to an $I_2$ embedding) and the critical point of an $I_2$ embedding is sometimes said to be $\omega$-fold superstrong. A further strengthening of $E_n$ is called $I_1$ or $E_\omega$. If for every $\gamma$ there is an $E_n$ embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be a P-$E_n$ cardinal; P-$E_0$ cardinals are also called $\omega$-fold extendible. If for every $\gamma$ there is an $I_2$ ($\omega$-fold superstrong) embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be $\omega$-fold strong (which is of course equivalent to P-$E_1$). Similiarly we can define P-$E_\omega$ cardinals.

• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma} \subset M$, $\kappa$ is said to be $\gamma$-strong. If $V_{j^{n-1}(\kappa+\gamma)} \subset M$, $\kappa$ is said to be $n$-fold $\gamma$-strong.
• If $\kappa$ is $\gamma$-strong for all $\gamma$, it is said to be strong. If $\kappa$ is $n$-fold $\gamma$-strong for all $\gamma$, it is said to be $n$-fold strong.
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ whose critical point, which we will call $\mu$, is less than $\kappa$ and such that $f"\mu \subset \mu$, $V_{j(f)(\mu)} \subset M$, and $j(f)(\mu)=f(\mu)$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be a Woodin cardinal. If for every such $f$ there are such $j: V \to M$ and $\mu$ such that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent), then $\kappa$ is said to be $n$-fold Woodin.
• If for every function $f: \kappa \to \kappa$ there is an measure $U_f \subset \mathcal{P}(\kappa)$ such that for $U_f$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{j(f)(\mu)} \subset M$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be weakly hyper-Woodin. One can define $n$-fold weakly hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent).
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$, then $\kappa$ is said to be a Shelah cardinal. If for every such $f$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(j^{n-1}(\kappa))} \subset M$, then $\kappa$ is said to be a $n$-fold Shelah cardinal.
• If there is an measure $U \subset \mathcal{P}(\kappa)$ such that for every function $f: \kappa \to \kappa$, $U$ satisfies that for $U$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{j(f)(\mu)} \subset M$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be hyper-Woodin. One can define $n$-fold hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent).
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. One can define $n$-fold high jump for strongness cardinals by requiring $V_{j^{n-1}(\theta)} \subset M$. For $n \ge 2$, unlike for $n=1$, $n$-fold high jump for strongness is not equivalent to $n$-fold superstrong but weaker than $n-1$-fold almost huge and stronger than $n$-fold super-high-jump.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$, then $\kappa$ is said to be superstrong. If $V_{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-superstrong.
• One can define $n$-fold globally superstrong as follows: for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$.
• If there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $\gamma$-extendible. If there is an elementary embedding $j: V_{j(\kappa+\gamma)} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $n$-fold $\gamma$-extendible.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^\lambda \subset M$, $\kappa$ is said to be $\lambda$-supercompact. If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^{n-1}(\lambda)} \subset M$, $\kappa$ is said to be $n$-fold $\lambda$-supercompact.
• If $\kappa$ is $\lambda$-supercompact for all $\gamma$, it is said to be supercompact. If $\kappa$ is $n$-fold $\lambda$-supercompact for all $\gamma$, it is said to be $n$-fold supercompact.
• If $\kappa$ is $\gamma$-extendible for all $\gamma$, it is said to be extendible. If $\kappa$ is $n$-fold $\gamma$-extendible for all $\gamma$, it is said to be $n$-fold extendible. Being $n$-fold extendible is equivalent to being $n+1$-fold strong.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $M^\theta \subset M$, $\kappa$ is said to be high jump. One can define $n$-fold high jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$, and clearance $\theta$ such that $M^\theta \subset M$ and $\theta \gt \gamma$, then $\kappa$ is said to be super-high-jump. One can define $n$-fold super-high-jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$ (and still $\theta \gt \gamma$).
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$, then $\kappa$ is said to be almost huge. If $M^{\lt j^n(\kappa)} \subset M$, $\kappa$ is said to be almost $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-huge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-$n$-huge
• An elementary embedding $j: V_{j(\kappa)} \to V_\eta$ with critical point $\kappa$ is called an $A_2$ embedding. The critical point can also be called a 2-fold 0-extendible cardinal. More generally, if $\kappa$ is the critical point of an elementary embedding $j: V_{j^{n-1}(\kappa)} \to V_\eta$, one can call it an $n$-fold 0-extendible cardinal.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$, then $\kappa$ is said to be huge. If $M^{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be superhuge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-$n$-huge
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $V_{j({\kappa+\gamma})} \subset M$, $\kappa$ is said to be $\gamma$-ultrahuge. One can define $n$-fold $\gamma$-ultrahuge by $M^{j^n(\kappa)} \subset M$ and $V_{\kappa+\gamma} \subset M$.
• If $\kappa$ is $\gamma$-ultrahuge for all $\gamma$, it is said to be ultrahuge. Thus one can define $n$-fold $\gamma$-ultrahuge to mean $n$-fold ultrahuge for all $\gamma$.
• 2-fold $\lambda$-supercompact cardinals are also called $\lambda$-hyperhuge and 2-fold supercompact cardinals are also called hyperhuge. Thus one can alternatively refer to $n+1$-fold $\lambda$-supercompact as $n$-fold $\lambda$-hyperhuge and $n+1$-fold supercompact cardinals as $n$-fold hyperhuge. For $n \ge 1$, a cardinal is $n$-fold hyperhuge iff it is $n+1$-fold extendible (thus iff it is $n+2$-fold strong).

• If there is a $\theta$ such that $V_\kappa$ is an elementary submodel of $V_\theta$, $\kappa$ is said to be 0-extendible or otherwordly; if additionally $\kappa$ is inaccessible, it is said to be 0-pseudo-uplifting; and if additionally $\kappa$ and $\theta$ are both inaccessible, it is said to be 0-uplifting. If there are unboundely many such $\theta$, $\kappa$ is said to be totally otherwordly, pseudo-uplifting or uplifting, respectively.
• If for every $A \subseteq V_\kappa$, there exists transitive models $M$ and $N$ and an elementary embedding $j: M \to N$ such that the critical point of $j$ is $\kappa$, $V_\kappa \subset M$, and $V_{j(\kappa)} \subset N$, then $\kappa$ is said to be weakly superstrong. If for every $\gamma$ and every $A \subseteq V_\kappa$ there exists such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be superstrongly unfoldable.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. If additionally $M^\theta \subset M$, $\kappa$ is said to be high jump. If for every $\gamma$ there is a high jump embedding $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be super-high-jump. One can similarly define super-high-jump for strongness cardinals but this property is equivalent to being globally superstrong just like high jump for strongness is equivalent to superstrong.
• The definitions of superstrong, almost huge and huge cardinals involve elementary embeddings $j: V \to M$ with critical point $\kappa$ and certain other properties. If for every $\gamma$ there is such a $j$ with $j(\kappa) \gt \gamma$, $\kappa$ is said to be globally superstrong, super-almost-huge, or superhuge, respectively. One can define a similar strengthening of the definition of $\gamma$-extendible cardinals (there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$); this strengthening doesn't appear to have a name but one could call such cardinals globally $\gamma$-extendible.
• An elementary embedding $j: V_\lambda \to V_\lambda$ is called a rank into rank, $I_3$ or $E_0$ embedding. Such an elementary embedding extends to a $\Sigma_0$-elementary embedding $\hat{j}: V_{\lambda+1} \to V_{\lambda+1}$; if $\hat{j}$ is $\Sigma_{2n}$-elementary (equivalently $\Sigma_{2n-1}$-elementary) it is called an $E_n$ embedding (for $n \lt \omega$). An elementary embedding $j: V \to M$ such that $j^n(\kappa) \subset M$ for all $n \lt \omega$ is called an $I_2$ embedding (any $I_2$ embedding restricts to an $E_1$ embedding and conversely any $E_1$ embedding extends to an $I_2$ embedding) and the critical point of an $I_2$ embedding is sometimes said to be $\omega$-fold superstrong. A further strengthening of $E_n$, involving a fully elementary embedding $j: V_{\lambda+1} \to V_{\lambda+1}$, is called $I_1$ or $E_\omega$. If for every $\gamma$ there is an $E_n$ embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be a P-$E_n$ cardinal; P-$E_0$ cardinals are also called $\omega$-fold extendible. If for every $\gamma$ there is an $I_2$ ($\omega$-fold superstrong) embedding with critical point $\kappa$ and $j(\kappa) \gt \gamma$, $\kappa$ is said to be $\omega$-fold strong (which is of course equivalent to P-$E_1$). Similiarly we can define P-$E_\omega$ cardinals.

• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma} \subset M$, $\kappa$ is said to be $\gamma$-strong. If $V_{j^{n-1}(\kappa+\gamma)} \subset M$, $\kappa$ is said to be $n$-fold $\gamma$-strong.
• If $\kappa$ is $\gamma$-strong for all $\gamma$, it is said to be strong. If $\kappa$ is $n$-fold $\gamma$-strong for all $\gamma$, it is said to be $n$-fold strong.
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ whose critical point, which we will call $\mu$, is less than $\kappa$ and such that $f"\mu \subset \mu$, $V_{j(f)(\mu)} \subset M$, and $j(f)(\mu)=f(\mu)$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be a Woodin cardinal. If for every such $f$ there are such $j: V \to M$ and $\mu$ such that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent), then $\kappa$ is said to be $n$-fold Woodin.
• If for every function $f: \kappa \to \kappa$ there is an measure $U_f \subset \mathcal{P}(\kappa)$ such that for $U_f$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{j(f)(\mu)} \subset M$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be weakly hyper-Woodin. One can define $n$-fold weakly hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent).
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$, then $\kappa$ is said to be a Shelah cardinal. If for every such $f$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(j^{n-1}(\kappa))} \subset M$, then $\kappa$ is said to be a $n$-fold Shelah cardinal.
• If there is an measure $U \subset \mathcal{P}(\kappa)$ such that for every function $f: \kappa \to \kappa$, $U$ satisfies that for $U$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{f(\mu)} \subset M$ and $j(f)(\mu)=f(\mu)$, then $\kappa$ is said to be hyper-Woodin (contrary to what I previously said, we can't drop the requirement that $j(f)(\mu)=f(\mu)$, even if we replace $V_{f(\mu)} \subset M$ with $V_{j(f)(\mu)} \subset M$). One can define $n$-fold hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{f(j^{n-1}(\mu))} \subset M$ and for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. One can define $n$-fold high jump for strongness cardinals by requiring $V_{j^{n-1}(\theta)} \subset M$. For $n \ge 2$, unlike for $n=1$, $n$-fold high jump for strongness is not equivalent to $n$-fold superstrong but weaker than $n-1$-fold almost huge and stronger than $n$-fold super-high-jump.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$, then $\kappa$ is said to be superstrong. If $V_{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-superstrong.
• One can define $n$-fold globally superstrong as follows: for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$.
• If there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $\gamma$-extendible. If there is an elementary embedding $j: V_{j(\kappa+\gamma)} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $n$-fold $\gamma$-extendible.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^\lambda \subset M$, $\kappa$ is said to be $\lambda$-supercompact. If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^{n-1}(\lambda)} \subset M$, $\kappa$ is said to be $n$-fold $\lambda$-supercompact.
• If $\kappa$ is $\lambda$-supercompact for all $\gamma$, it is said to be supercompact. If $\kappa$ is $n$-fold $\lambda$-supercompact for all $\gamma$, it is said to be $n$-fold supercompact.
• If $\kappa$ is $\gamma$-extendible for all $\gamma$, it is said to be extendible. If $\kappa$ is $n$-fold $\gamma$-extendible for all $\gamma$, it is said to be $n$-fold extendible. Being $n$-fold extendible is equivalent to being $n+1$-fold strong.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $M^\theta \subset M$, $\kappa$ is said to be high jump. One can define $n$-fold high jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$, and clearance $\theta$ such that $M^\theta \subset M$ and $\theta \gt \gamma$, then $\kappa$ is said to be super-high-jump. One can define $n$-fold super-high-jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$ (and still $\theta \gt \gamma$).
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$, then $\kappa$ is said to be almost huge. If $M^{\lt j^n(\kappa)} \subset M$, $\kappa$ is said to be almost $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-huge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-$n$-huge
• An elementary embedding $j: V_{j(\kappa)} \to V_\eta$ with critical point $\kappa$ is called an $A_2$ embedding. The critical point can also be called a 2-fold 0-extendible cardinal. More generally, if $\kappa$ is the critical point of an elementary embedding $j: V_{j^{n-1}(\kappa)} \to V_\eta$, one can call it an $n$-fold 0-extendible cardinal.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$, then $\kappa$ is said to be huge. If $M^{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be superhuge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-$n$-huge
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $V_{j({\kappa+\gamma})} \subset M$, $\kappa$ is said to be $\gamma$-ultrahuge. One can define $n$-fold $\gamma$-ultrahuge by $M^{j^n(\kappa)} \subset M$ and $V_{\kappa+\gamma} \subset M$.
• If $\kappa$ is $\gamma$-ultrahuge for all $\gamma$, it is said to be ultrahuge. Thus one can define $n$-fold $\gamma$-ultrahuge to mean $n$-fold ultrahuge for all $\gamma$.
• 2-fold $\lambda$-supercompact cardinals are also called $\lambda$-hyperhuge and 2-fold supercompact cardinals are also called hyperhuge. Thus one can alternatively refer to $n+1$-fold $\lambda$-supercompact as $n$-fold $\lambda$-hyperhuge and $n+1$-fold supercompact cardinals as $n$-fold hyperhuge. For $n \ge 1$, a cardinal is $n$-fold hyperhuge iff it is $n+1$-fold extendible (thus iff it is $n+2$-fold strong).

If a family of large cardinal notions is defined with an ordinal parameter (call it $\gamma-A$), one can define the property of being $\gamma-A$ for all $\gamma$:

If the definition of a large cardinal notion $A(\kappa)$ asserts the existence of a cardinal $\theta \gt \kappa$, one can define a large cardinal notion asserting that there are unboundedly many such $\theta$, that is, for every ordinal $\gamma$ there is such a $\theta$ (is that what you mean by superfication?):

If the definition of a large cardinal notion $A(\kappa)$ asserts the existence of a cardinal $\theta \gt \kappa$, one can define a $C^{(n)}$ variant, additionally asserting that $\theta$ is $\Sigma_n$-correct.

• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma} \subset M$, $\kappa$ is said to be $\gamma$-strong. If $V_{j^{n-1}(\kappa+\gamma)} \subset M$, $\kappa$ is said to be $n$-fold $\gamma$-strong.
• If $\kappa$ is $\gamma$-strong for all $\gamma$, it is said to be strong. If $\kappa$ is $n$-fold $\gamma$-strong for all $\gamma$, it is said to be $n$-fold strong.
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ whose critical point, which we will call $\mu$, is less than $\kappa$ and such that $f"\mu \subset \mu$, $V_{j(f)(\mu)} \subset M$, and $j(f)(\mu)=f(\mu)$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be a Woodin cardinal. If for every such $f$ there are such $j: V \to M$ and $\mu$ such that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent), then $\kappa$ is said to be $n$-fold Woodin.
• If for every function $f: \kappa \to \kappa$ there is an measure $U_f \subset \mathcal{P}(\kappa)$ such that for $U_f$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{j(f)(\mu)} \subset M$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be weakly hyper-Woodin. One can define $n$-fold weakly hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent).
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$, then $\kappa$ is said to be a Shelah cardinal. If for every such $f$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(j^{n-1}(\kappa))} \subset M$, then $\kappa$ is said to be a $n$-fold Shelah cardinal.
• If there is an measure $U_f \subset \mathcal{P}(\kappa)$ such that for every function $f: \kappa \to \kappa$, $U$ satisfies that for $U$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{j(f)(\mu)} \subset M$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be hyper-Woodin. One can define $n$-fold hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent).
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. One can define $n$-fold high jump for strongness cardinals by requiring $V_{j^{n-1}(\theta)} \subset M$. For $n \ge 2$, unlike for $n=1$, $n$-fold high jump for strongness is not equivalent to $n$-fold superstrong but weaker than $n-1$-fold almost huge and stronger than $n$-fold super-high-jump.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$, then $\kappa$ is said to be superstrong. If $V_{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-superstrong.
• One can define $n$-fold globally superstrong as follows: for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$.
• If there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $\gamma$-extendible. If there is an elementary embedding $j: V_{j(\kappa+\gamma)} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $n$-fold $\gamma$-extendible.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^\lambda \subset M$, $\kappa$ is said to be $\lambda$-supercompact. If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^{n-1}(\lambda)} \subset M$, $\kappa$ is said to be $n$-fold $\lambda$-supercompact.
• If $\kappa$ is $\lambda$-supercompact for all $\gamma$, it is said to be supercompact. If $\kappa$ is $n$-fold $\lambda$-supercompact for all $\gamma$, it is said to be $n$-fold supercompact.
• If $\kappa$ is $\gamma$-extendible for all $\gamma$, it is said to be extendible. If $\kappa$ is $n$-fold $\gamma$-extendible for all $\gamma$, it is said to be $n$-fold extendible. Being $n$-fold extendible is equivalent to being $n+1$-fold strong.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $M^\theta \subset M$, $\kappa$ is said to be high jump. One can define $n$-fold high jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$, and clearance $\theta$ such that $M^\theta \subset M$ and $\theta \gt \gamma$, then $\kappa$ is said to be super-high-jump. One can define $n$-fold super-high-jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$ (and still $\theta \gt \gamma$).
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$, then $\kappa$ is said to be almost huge. If $M^{\lt j^n(\kappa)} \subset M$, $\kappa$ is said to be almost $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-huge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-$n$-huge
• An elementary embedding $j: V_{j(\kappa)} \to V_\eta$ with critical point $\kappa$ is called an $A_2$ embedding. The critical point can also be called a 2-fold 0-extendible cardinal. More generally, if $\kappa$ is the critical point of an elementary embedding $j: V_{j^{n-1}(\kappa)} \to V_\eta$, one can call it an $n$-fold 0-extendible cardinal.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$, then $\kappa$ is said to be huge. If $M^{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be superhuge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-$n$-huge
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $V_{j({\kappa+\gamma})} \subset M$, $\kappa$ is said to be $\gamma$-ultrahuge. One can define $n$-fold $\gamma$-ultrahuge by $M^{j^n(\kappa)} \subset M$ and $V_{\kappa+\gamma} \subset M$.
• If $\kappa$ is $\gamma$-ultrahuge for all $\gamma$, it is said to be ultrahuge. Thus one can define $n$-fold $\gamma$-ultrahuge to mean $n$-fold ultrahuge for all $\gamma$.
• 2-fold $\lambda$-supercompact cardinals are also called $\lambda$-hyperhuge and 2-fold supercompact cardinals are also called hyperhuge. Thus one can alternatively refer to $n+1$-fold $\lambda$-supercompact as $n$-fold $\lambda$-hyperhuge and $n+1$-fold supercompact cardinals as $n$-fold hyperhuge. For $n \ge 1$, a cardinal is $n$-fold hyperhuge iff it is $n+1$-fold extendible (thus iff it is $n+2$-fold strong).

• If one thus defines ($n$-fold) Woodin for extendibility cardinals in analogy with ($n$-fold) Woodin cardinals, one gets a simplified definition of ($n$-fold) Vopenka cardinals; $n$-fold Woodin for extendibility/$n$-fold Vopenka is equivalent to $n+1$-fold Woodin and to $n$-fold Woodin for supercompactness.
• In the same way, one can define ($n$-fold) Shelah for extendibility cardinals in analogy with ($n$-fold) Shelah cardinals; $n$-fold Shelah for extendibility is equivalent to $n+1$-fold Shelah and to $n$-fold Shelah for supercompactness.
• One can define ($n$-fold) high jump for extendibility cardinals in analogy with high jump cardinals; $n$-fold high jump for extendibility is weaker than $n$-fold high jump but stronger than $n+1$-fold hyper-Woodin.
• One can define ($n$-fold) hyper-Woodin for extendibility or ($n$-fold) hyper-Vopenka cardinals in analogy with ($n$-fold) hyper-Woodin cardinals; $n$-fold hyper-Vopenka is equivalent to $n+1$-fold hyper-Woodin and to $n+1$-fold hyper-Woodin for supercompactness.
• One can define ($n$-fold) weakly hyper-Woodin for extendibility or ($n$-fold) weakly hyper-Vopenka cardinals in analogy with ($n$-fold) weakly hyper-Woodin cardinals; $n$-fold weakly hyper-Vopenka is equivalent to $n+1$-fold weakly hyper-Woodin and to $n+1$-fold weakly hyper-Woodin for supercompactness.

If a family of large cardinal notions is defined with an ordinal parameter (call it $\gamma$-$\phi$), one can define the property of being $\gamma$-$\phi$ for all $\gamma$:

If the definition of a large cardinal notion $\phi(\kappa)$ asserts the existence of a cardinal $\theta \gt \kappa$, one can define a large cardinal notion asserting that there are unboundedly many such $\theta$, that is, for every ordinal $\gamma$ there is such a $\theta$ (is that what you mean by superfication?):

If the definition of a large cardinal notion $\phi(\kappa)$ asserts the existence of a cardinal $\theta \gt \kappa$, one can define a $C^{(n)}$ variant, additionally asserting that $\theta$ is $\Sigma_n$-correct.

• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma} \subset M$, $\kappa$ is said to be $\gamma$-strong. If $V_{j^{n-1}(\kappa+\gamma)} \subset M$, $\kappa$ is said to be $n$-fold $\gamma$-strong.
• If $\kappa$ is $\gamma$-strong for all $\gamma$, it is said to be strong. If $\kappa$ is $n$-fold $\gamma$-strong for all $\gamma$, it is said to be $n$-fold strong.
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ whose critical point, which we will call $\mu$, is less than $\kappa$ and such that $f"\mu \subset \mu$, $V_{j(f)(\mu)} \subset M$, and $j(f)(\mu)=f(\mu)$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be a Woodin cardinal. If for every such $f$ there are such $j: V \to M$ and $\mu$ such that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent), then $\kappa$ is said to be $n$-fold Woodin.
• If for every function $f: \kappa \to \kappa$ there is an measure $U_f \subset \mathcal{P}(\kappa)$ such that for $U_f$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{j(f)(\mu)} \subset M$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be weakly hyper-Woodin. One can define $n$-fold weakly hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent).
• If for every function $f: \kappa \to \kappa$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(\kappa)} \subset M$, then $\kappa$ is said to be a Shelah cardinal. If for every such $f$ there is an elementary embedding $j: V \to M$ such that $V_{j(f)(j^{n-1}(\kappa))} \subset M$, then $\kappa$ is said to be a $n$-fold Shelah cardinal.
• If there is an measure $U \subset \mathcal{P}(\kappa)$ such that for every function $f: \kappa \to \kappa$, $U$ satisfies that for $U$-almost all $\mu \lt \kappa$, $f"\mu \subset \mu$ and there is an elementary embedding $j: V \to M$ with critical point $\mu$ such that $V_{j(f)(\mu)} \subset M$ (it is provable that a definition that additionally requires $j(f)(\mu)=f(\mu)$ is equivalent), then $\kappa$ is said to be hyper-Woodin. One can define $n$-fold hyper-Woodin cardinals by requiring for witnessing embeddings $j$ that $V_{j^n(f)(j^{n-1}(\mu))} \subset M$ (it is provable that a definition that additionally requires that for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$ is equivalent).
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $V_\theta \subset M$, $\kappa$ is said to be high jump for strongness. One can define $n$-fold high jump for strongness cardinals by requiring $V_{j^{n-1}(\theta)} \subset M$. For $n \ge 2$, unlike for $n=1$, $n$-fold high jump for strongness is not equivalent to $n$-fold superstrong but weaker than $n-1$-fold almost huge and stronger than $n$-fold super-high-jump.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$, then $\kappa$ is said to be superstrong. If $V_{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-superstrong.
• One can define $n$-fold globally superstrong as follows: for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$.
• If there is an elementary embedding $j: V_{\kappa+\gamma} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $\gamma$-extendible. If there is an elementary embedding $j: V_{j(\kappa+\gamma)} \to V_\eta$ with critical point $\kappa$, $\kappa$ is said to be $n$-fold $\gamma$-extendible.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^\lambda \subset M$, $\kappa$ is said to be $\lambda$-supercompact. If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^{n-1}(\lambda)} \subset M$, $\kappa$ is said to be $n$-fold $\lambda$-supercompact.
• If $\kappa$ is $\lambda$-supercompact for all $\gamma$, it is said to be supercompact. If $\kappa$ is $n$-fold $\lambda$-supercompact for all $\gamma$, it is said to be $n$-fold supercompact.
• If $\kappa$ is $\gamma$-extendible for all $\gamma$, it is said to be extendible. If $\kappa$ is $n$-fold $\gamma$-extendible for all $\gamma$, it is said to be $n$-fold extendible. Being $n$-fold extendible is equivalent to being $n+1$-fold strong.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$, whose clearance we will call $\theta$, such that $M^\theta \subset M$, $\kappa$ is said to be high jump. One can define $n$-fold high jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$, and clearance $\theta$ such that $M^\theta \subset M$ and $\theta \gt \gamma$, then $\kappa$ is said to be super-high-jump. One can define $n$-fold super-high-jump cardinals by requiring $M^{j^{n-1}(\theta)} \subset M$ (and still $\theta \gt \gamma$).
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$, then $\kappa$ is said to be almost huge. If $M^{\lt j^n(\kappa)} \subset M$, $\kappa$ is said to be almost $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-huge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{\lt j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-almost-$n$-huge
• An elementary embedding $j: V_{j(\kappa)} \to V_\eta$ with critical point $\kappa$ is called an $A_2$ embedding. The critical point can also be called a 2-fold 0-extendible cardinal. More generally, if $\kappa$ is the critical point of an elementary embedding $j: V_{j^{n-1}(\kappa)} \to V_\eta$, one can call it an $n$-fold 0-extendible cardinal.
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$, then $\kappa$ is said to be huge. If $M^{j^n(\kappa)} \subset M$, $\kappa$ is said to be $n$-huge.
• If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be superhuge. If for every $\gamma$ there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j^n(\kappa)} \subset M$ and $j(\kappa) \gt \gamma$, then $\kappa$ is said to be super-$n$-huge
• If there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $M^{j(\kappa)} \subset M$ and $V_{j({\kappa+\gamma})} \subset M$, $\kappa$ is said to be $\gamma$-ultrahuge. One can define $n$-fold $\gamma$-ultrahuge by $M^{j^n(\kappa)} \subset M$ and $V_{\kappa+\gamma} \subset M$.
• If $\kappa$ is $\gamma$-ultrahuge for all $\gamma$, it is said to be ultrahuge. Thus one can define $n$-fold $\gamma$-ultrahuge to mean $n$-fold ultrahuge for all $\gamma$.
• 2-fold $\lambda$-supercompact cardinals are also called $\lambda$-hyperhuge and 2-fold supercompact cardinals are also called hyperhuge. Thus one can alternatively refer to $n+1$-fold $\lambda$-supercompact as $n$-fold $\lambda$-hyperhuge and $n+1$-fold supercompact cardinals as $n$-fold hyperhuge. For $n \ge 1$, a cardinal is $n$-fold hyperhuge iff it is $n+1$-fold extendible (thus iff it is $n+2$-fold strong).

• If one thus defines ($n$-fold) Woodin for extendibility cardinals in analogy with ($n$-fold) Woodin cardinals, one gets a simplified definition of ($n$-fold) Vopenka cardinals; $n$-fold Woodin for extendibility/$n$-fold Vopenka is equivalent to $n+1$-fold Woodin and to $n$-fold Woodin for supercompactness.
• In the same way, one can define ($n$-fold) Shelah for extendibility cardinals in analogy with ($n$-fold) Shelah cardinals; $n$-fold Shelah for extendibility is equivalent to $n+1$-fold Shelah and to $n$-fold Shelah for supercompactness.
• One can define ($n$-fold) high jump for extendibility cardinals in analogy with high jump cardinals; $n$-fold high jump for extendibility is weaker than $n$-fold high jump but stronger than $n+1$-fold hyper-Woodin.
• One can define ($n$-fold) hyper-Woodin for extendibility or ($n$-fold) hyper-Vopenka cardinals in analogy with ($n$-fold) hyper-Woodin cardinals; $n$-fold hyper-Vopenka is equivalent to $n+1$-fold hyper-Woodin and to $n+1$-fold hyper-Woodin for supercompactness.
• One can define ($n$-fold) weakly hyper-Woodin for extendibility or ($n$-fold) weakly hyper-Vopenka cardinals in analogy with ($n$-fold) weakly hyper-Woodin cardinals; $n$-fold weakly hyper-Vopenka is equivalent to $n+1$-fold weakly hyper-Woodin and to $n$-fold weakly hyper-Woodin for supercompactness.