It seems (to me, at least) natural to model a position in a game as the set of all positions that can be reached in one move. But then well-foundedness prevents you from using this representation to represent games with loops.

If you're interested in modelling connectivity in web pages, it seems natural to me to represent a web page as the set of web pages it links to. But links form cycles.

Computer science abounds with examples of types that can be awkward to model with well-founded sets. For example, in many computing environments elements of lists are represented by pointers to a block of data, not by the block of data itself. So it's often straightforward to construct a list, say, that contains itself, or at least contains loops of inclusions, because under the hood the self-containment is indirect. It'd be nice to model these with non-well founded sets because you'd like to abstract away from the fact that containment is implemented indirectly so you can reason directly about containment.