Can anybody give me an example of a "naturally-occurring" algebraic category C in which:

1. C has two non-isomorphic objects A and B which are bi-embeddable via monic maps; but

2. C does NOT have any infinite collection A_1, A_2, ... of objects which are pairwise bi-embeddable (via monic maps) and pairwise nonisomorphic?

Alternatively: can anybody give a reason why, under some reasonable hypothesis about the category C, property 1 should imply that 2 fails ("there is an infinite collection of pairwise bi-embeddable, pairwise nonisomorphic objects")?