Natural example: the category of function fields of supersingular elliptic curves over a fixed algebraically closed field of characteristic $p$ (you pick $p$, say bigger than $13$ or so to ensure more than one isomorphism class).  Note that this is -- up to translations -- the opposite category to the Brandt module category, in which the objects are the ss elliptic curves themselves and the morphisms are isogenies.

As $p$ varies, this gives a family of (essentially) finite categories such that any two objects are mutually embeddable and the number of isomorphism classes of objects tends to infinity with $p$.

If I may be so bold, I spent much of a paper talking about the relation between two function fields that each is embeddable in the other, which I called (borrowing from the theory of elliptic curves) "isogeny".  

>PLC, On elementary equivalence, isomorphism and isogeny.  
J. Théor. Nombres Bordeaux 18 (2006), no. 1, 29--58.

http://alpha.math.uga.edu/~pete/logic.pdf