A monic epi that is not an iso
Consider the ring map It is an epimorphism because is initial in Ring. It is a monomorphism because maps into that agree after composition with agree everywhere.
Another example would be the inclusion in the category Haus. This follows from the density of in .
Jargon. A category where all monic epis are isos is called balanced. A famous (and pleasant!) property of toposes is that all toposes are balanced.