A common trend in large cardinal axioms dealing with critical points of elementary embeddings from $V$ into a transitive class $M$ is to make some large cardinal axiom with an ordinal parameter, similar to "$M$ is closed under $\theta$-sequences" or "$V_\theta$ is a subset of $M$". The resulting cardinals from these definitions are the $\theta$-supercompacts and the $\theta$-strongs.
After this, expansions could be made to make the cardinal even larger; specifically, by taking that parameter of $M$ and making it $j^n(\kappa)$. When doing this to $\theta$-strong cardinals, one achieves $n$-superstrong cardinals. When doing this to supercompact cardinals, one achieves $n$-huge cardinals.
Of course, a natural question at this point is to see what happens when one does it to $\eta$-extendible cardinals. Here are the results:
Let $\kappa$ be $n$-superextendible if there is some ordinal $\eta$ such that $\kappa$ is the critical point of some nontrivial elementary embedding $j:V_\theta\prec V_\eta$ such that $\theta>j^n(\kappa)$.
So far, here are the results:
- The $0$-superextendible cardinals are precisely the $0$-extendible cardinals
- I3 cardinals are $n$-superextendible for every $n$ (this property could be called $\omega$-superextendible)
- The $1$-superextendible cardinals, if one lets an arbitrary such be $\kappa$, are $\kappa$-extendible. Of course, every $1$-superextendible cardinal is a supercompact cardinal and is, in fact, a stationary limit of such due to this property.
- If a cardinal is $n+1$-superextendible it is $n$-superextendible
These properties are, yes, very obvious. Are there any other possibly less obvious properties that I am missing? Where do these cardinals fall under consistency strength?