Is there a "large powerset axiom" so extreme that it disproves the existence of strongly inaccessible cardinals? 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 raises 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 $\neg \mathrm{Con}(ZFC + \varphi).$

 A: Let me add another example, which is more known.
Consider the following:

Tree property holds at all regular cardinals $\geq \aleph_2.$ 

If this statement is consistent is a well-know question of Magidor  (1970$^{th}$), and is more famous than Foreman's maximality principle. There are some results supporting this principle. It also implies:
1) $GCH$ fails everywhere (if $2^\kappa=\kappa^+$, then there is a special $\kappa^{++}$-Aronszajn tree),
2) There are no inaccessible cardinals (if $\kappa$ is inaccessible, then there is a special $\kappa^+$-Aronszajn tree).
A: Foreman's maximality principle is as you have requested, though it is not yet known if it is consistent or not.
Foreman's maximality principle: Any non-trivial forcing notion either it adds a real or colapses some cardinals.
It follows from it that:
1) $GCH$ fails everywhere,
2) there are no inaccessible cardinals.
This principle is stated in the following paper:
Foreman, Magidor, Shelah, "$0^\sharp$  and some forcing principles",  J. Symbolic Logic, 51 (1986) 39-46.
See also "Questions about $\aleph_1-$closed forcing notions".
