Is ZFC + the following statement consistent (and if so, is it equiconsistent to some known large cardinal):
For every function $f:ORD \rightarrow ORD$ such that:
$f(\alpha)\geq \alpha$,
$\alpha > \beta \Rightarrow f(\alpha) \geq f(\beta)$
$\{\kappa|\kappa \text{ is regular} \wedge \forall \alpha <\kappa (f(\alpha)<\kappa)\}$ is a proper class.
This is obviously not a theorem of ZFC, as we can take $f(\alpha)=|2^{\aleph_\alpha}|$ and conclude that we have a proper class of inaccessible cardinals. Actually, using $f(\alpha) = \alpha\text{-th-inaccessible cardinal}$ we get 1-inaccessible cardinals, and then we can get hyper-inaccessibles and get at least as far up as Mahlo cardinals, maybe more.
So either this statement is somehow plainly inconsistent, or it has a consistency strength of some large cardinal.
Note: the idea behind this construction is to "take away the uniqueness" of the power function in constructing the inaccessibles. Literally, what's 'the most' we can get out of a particular function.