A generic monomorphism is a monomorphism of which every monomorphisms in the same category is a pullback; it is like a subobject classifier without uniqueness. I am looking for a locally Cartesian closed regular category $C$ whose ex/reg completion is a topos, even though $C$ has no generic monomorphism.
Matias Menni proved that if a regular category is locally Cartesian closed and has a generic monomorphism, then its ex/reg completion is a topos. The ex/reg completion adds quotients objects to a category in a way that preserves all regular epimorphisms (in contrast to the exact or ex/lex completion, which only preserves split epimorphisms). I suspect that the converse is false, but I can't find any counterexamples.