I know several ways to modify large cardinal notions:

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 $\kappa$ is *$\gamma$-shrewd* for all $\gamma$, it is said to be *shrewd*.
 - If $\kappa$ is *$\gamma$-strongly unfoldable* for all $\gamma$, it is said to be *strongly unfoldable* (equivalent to being shrewd).
 - If $\kappa$ is *$\gamma$-strong* for all $\gamma$, it is said to be *strong*.
 - If $\kappa$ is *$\lambda$-supercompact* for all $\lambda$, it is said to be *supercompact*.
 - If $\kappa$ is *$\gamma$-extendible* for all $\gamma$, it is said to be *extendible*.

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 $\theta$:

 - If there is a $\theta$ such that $V_\kappa$ is an elementary submodel of $V_\theta$, $\kappa$ is said to be *[0-extendible](https://arxiv.org/abs/1307.3486)* or *[otherwordly](https://jdh.hamkins.org/otherwordly-cardinals/)*; 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](https://arxiv.org/abs/2107.01580)*, *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](https://doi.org/10.1016/j.apal.2007.02.003)*. 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](https://doi.org/10.1016/j.apal.2007.02.003); $P-E_0$ 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 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 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*, *almost huge* and *huge* 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.
 - 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.
 - 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*, and *$C^{(n)}$-superhuge* cardinals.

For most large cardinal notions stronger than measurable and weaker than wholeness axioms, one can define [$n$-fold variants](https://doi.org/10.1016/j.apal.2007.02.003):

 - 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 $f"\mu \subset \mu$, $V_{f(\mu)} \subset M$, and $j(f)(\mu)=f(\mu)$, 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 $V_{f(j^{n-1}(\mu))} \subset M$, $V_{f(\kappa)} \subset M$ and for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$, 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 $V_{f(\mu)} \subset M$ and $j(f)(\mu)=f(\mu)$, 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 $V_{f(j^{n-1}(\mu))} \subset M$ and, for every $\alpha \le j^{n-1}(\mu)$, $j(f)(\alpha)=f(\alpha)$.
 - 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 $V_{f(\mu)} \subset M$ and $j(f)(\mu)=f(\mu)$, then $\kappa$ is said to be *hyper-Woodin*. 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.
 - 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 *[$A_2$](https://doi.org/10.1016/0003-4843(78)90031-1)* 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](https://arxiv.org/abs/1708.07561)* and 2-fold supercompact cardinals are also called *[hyperhuge](https://www.worldscientific.com/doi/abs/10.1142/S021906131750009X)*. 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).

One can also define new large cardinal notions by replacing strongness embeddings ($j: V \to M$ with $V_\zeta \subset M$) by supercompactness embeddings ($j: V \to M$ with $M^\lambda \subset M$):

 - *($n$-fold) Woodin for supercompactness* cardinals are defined in analogy with ($n$-fold) Woodin cardinals; $n$-fold Woodin for supercompactness is equivalent to $n+1$-fold Woodin.
 - *($n$-fold) Shelah for supercompactness* cardinals are defined in analogy with ($n$-fold) Shelah cardinals; $n$-fold Shelah for supercompactness is equivalent to $n+1$-fold Shelah.
 - One can also define *($n$-fold) weakly hyper-Woodin for supercompactness* cardinals in analogy with ($n$-fold) weakly hyper-Woodin cardinals; $n$-fold weakly hyper-Woodin for supercompactness is equivalent to $n+1$-fold weakly hyper-Woodin.
 - One can define *($n$-fold) hyper-Woodin for supercompactness* cardinals in analogy with ($n$-fold) hyper-Woodin cardinals; $n$-fold hyper-Woodin for supercompactness is equivalent to $n+1$-fold hyper-Woodin.

Similarly, one can define new large cardinal notions by replacing supercompactness embeddings by strongness embeddings:

 - *High jump for strongness* cardinals are defined in analogy with high jump cardinals.
 - One can in the same way define *($n$-fold) super-high-jump for strongness* in analogy with ($n$-fold) super-high-jump cardinals.

Additionally, one can define new large cardinal notions by replacing strongness or supercompactness embeddings by extendibility embeddings ($j: V_\zeta \to V_\eta$):

 - 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.