Is it consistent with Morse-Kelley set theory without global choice (but with choice for sets) that there are $2^{|V|}$ proper classes of different cardinalities?
Alternative question: Is it consistent with ZF that there is an inaccessible $κ$ such that $V_κ⊨\text{ZFC}$ and there are at least $|V_{κ+1}|$ cardinalities of subsets of $V_κ$?
With global choice, all proper classes are equinumerous with $V$, so the question is whether without global choice the extreme opposite can hold.
Morse-Kelley set theory (MK) has full comprehension for proper classes. One formalization of $2^{|V|}$ cardinalities of classes is existence of a formula (allowing parameters) encoding a function $F$ (with $\operatorname{ran}(F)⊆\operatorname{dom}(F)=P(V)$) such that $∀A \, ∀B \, (A≠B ⇒|F(A)|≠|F(B)|)$. Inconsistency would then be a schema over formulas. However, this formalization has a defect of (in a sense) requiring definability, hence the inclusion of the alternative question above.
I suspect that the answer is yes, even if having different cardinalities is strengthened to incomparability under surjections. Also, for first-order-definable (with set parameters) proper classes, I suspect that having $|V|$ different cardinalities is consistent (for a partial result, see here).
