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 $\gamma$-supercompact for all $\gamma$, 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*.
 - The definitions of *superstrong*, *high jump*, *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-high-jump*, *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.

*To be continued.*