"Bootstrapping" an unbounded class of inaccessible cardinals

$\kappa$ is superhuge if for any $\gamma$, there exists $j: V\to M$ such that $$crit(j)=\kappa,$$ $$\gamma<j(\kappa)$$ and $${}^{j(\kappa)}M\subset M.$$ But $j(\kappa)$ (inaccessible in M) must be inaccessible in $V$ as well.

If $\kappa$ is extendible, then there exists a proper class of inaccessible cardinals (and even more).

The maximality principle implies a sweeping uniform version of the idea you mention.

• Hamkins, Joel David, A simple maximality principle., J. Symb. Log. 68, No. 2, 527-550 (2003). ZBL1056.03028.

Namely, theorem 14 of that paper shows that if MP holds, then there is a proper class of inaccessible cardinals, if any; and similarly for Mahlo cardinals and indeed any large cardinal notion that is downward absolute to small-forcing grounds.

To explain, the maximality principle is the principle that any statement that is forceably necessary is already true. That is, if there is a forcing extension such that the statement is true in all further forcing extensions, then the statement is already true. This principle is expressible in modal terms by the scheme $\Diamond\Box\varphi\to\varphi$, and indeed, the modal logic of forcing was introduced in this paper specifically because of the connection with the maximality principle.

The point now is that if MP holds and there is an inaccessible cardinal, then there cannot be only one or a bounded number of them, because then we could collapse them all and thereby make it forcing necessarily true that there would be none; so under MP there would have to have been none to begin with. So if there is one, there must be a proper class of them.