While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms tend to be somewhat arbitrary (e.g. adding an axiom that there are no inaccessable cardinals, or that there is exactly one inaccessible cardinal). In particular, the examples from the referenced question/answer place an arbitrary limit on the height of the cumulative hierarchy, where any one of the suggested 'new' axioms could just be replaced by another axiom specifying an even larger inaccessable cardinal.

So I'm wondering if it can be shown that there is a 'largest' axiomatization, in the sense that no categorical extension of ZFC2 has a larger model (or if the opposite can be shown i.e. that there is no largest)?

Of course, if we place arbitrary restrictions on large cardinals in our meta-theory, we could trivially make this question true (e.g. by assuming there is exactly one inaccessible in the meta-theory). A reasonable set of metatheory assumptions for the purpose of this question would be ZFC + large cardinal axioms i.e. assuming that relevant large cardinals *do* exist (if necessary). I'm assuming standard (e.g. non-Henkin) semantics for 2nd order logic.

Also, my understanding is that the 2nd order Lowenheim number does at least provide an upper bound on the size of the models associated with categorical extensions of ZFC2 [2], and per this question [3], it sounds like the Lowenheim bound would not be any larger for nth order logic compared to 2nd order logic.

[2] http://www.logic.math.helsinki.fi/people/jouko.vaananen/JV96.pdf