The possibility of monic epics that are not isomorphisms does not survive any strengthening of “monic” or “epic.” Any monic extremal epimorphism is necessarily an isomorphism, and therefore so is any monic strong epimorphism or regular epimorphism (and dually). It follows that if all epics, or all monos, are extremal, then the category is automatically balanced.
In an “unbalanced” category it is frequently the case that the monomorphisms, the epimorphisms, or both, are not the “right” notion to consider and should be replaced by their extremal, strong, or regular counterparts.
Some algebraic categories, such as the category of groups, are balanced.
Any abelian category is balanced.
The category of rings is not balanced; is monic and epic but not an isomorphism.