If $\kappa$ is a strongly compact cardinal, then the singular cardinal hypothesis holds above $\kappa$. Hence the existence of large cardinals at the level of "strongly compact" or above is incompatible with even (apparently) mild large powerset axioms like "$2^\kappa$ always strictly exceeds $\kappa^+.$"

This begs the question:

Is there a "large powerset axiom" $\varphi$ so extreme that $\mathrm{ZFC}+\varphi$ disproves the existence of strongly inaccessible cardinals? Let us also require that $\mathrm{ZFC}+\varphi$ does not prove $\neg \mathrm{Con}(ZFC + \varphi).$