By an "inner model," let us mean a transitive subclass of the universe satisfying $\mathrm{ZFC}.$ Given that, here's an (intentionally) vague definition.
Definition. Call a set of axioms $\Phi$ in the language of set theory limiting iff there exists a (consistent, as far as we know) large cardinal axiom $\lambda$ such that no model of $\mathrm{ZFC}+\Phi$ has an inner model satisfying $\lambda$.
Example. $\{V=L\}$ is limiting, since (if I recall correctly) it contradicts the existence of $ω_1$-Erdős cardinals, and implies that the universe has no proper inner models.
Anyway, here's an (intentionally) vague question.
Question. Let $\Phi$ denote the statement that the generalized continuum hypothesis holds, and that no inaccessible cardinals exist. Is $\Phi$ limiting?