The nLab lists a bunch of examples for internal categories in various categories. If we think of a topos as a "universe" for mathematics the need for internal categories in a topos becomes obvious.
How about if the ambient category is not a topos? There are some examples (given on the linked page), but there are not familiar to me and more importantly I don't see how there are "natural", i.e. how you get interested in them without already knowing internal categories. Hence the question:
Is there an obvious application of internal category theory outside from topoi?
I'd like to see an example, which can be understood by a typical undergraduate who knows a bit of category theory, but no internal category theory, if that is at all possible.