Modal logics axiomatised with nesting depth one which are unlikely to be in PSPACE?
I just realized that there is a nice solution to your problem, if you are willing to consider linear logic as your ambient logic instead of intuitionistic or classical logic. As is well-known, linear logic with the exponential modality $!A$ is not decidable. Furthermore, the exponential is a comonad featuring the duplication axiom $!A \multimap !!A$, which is evidently an axiom of nesting depth 2. 

(I got this far right away, and then I got stuck -- which is why this answer is so late.)

However, I just realized that in implicit complexity, people modify the exponential $!A$ of linear logic to more precisely control the space and time usage of cut-elimination. Critically, all systems for doing so eliminate the duplication axiom! As a result, you can choose a system for which normalization likely goes past PSPACE (e.g., Elementary Affine Logic is as strong as elementary bounded Turing machines), and then the axiomatization of that will be unlikely to be in PSPACE, since that would imply that you could find cut-free proofs quickly. 

Link: Ugo dal Lago and Simone Martini, Phase Semantics and Decidability of Elementary Affine Logic