First of all, it handles the "get me the 2nd to last element" case automatically, but in a way that doesn't feel like a weird edge case: it's more "mathematically sound", basically. I always want mathematical soundness if possible because it leads to serendipity, the opposite of technical debt. Where technical debt is "dammit, this is going to take so much longer than it should!"; serendipity is "oh wow I can implement this cool new feature just by combining these other two things in a new way, in like 2 lines. This is going to be way faster than I thought." Mathematical soundness / purity leads to serendipity.
Directly, it supports caches very well. You just increment the number of things you've ever cached and that's where your next cached value goes; you don't care when it overwrites an old value.
There are other cases where you just need some variant of a thing, but you don't actually care that much about which variant you get. You might want to vary your wording in auto-generated text, for instance, by rotating synonyms. Or rotating the tiles you use in a 2D game. In this case I'd define an interface where you pass in a "seed" integer and it gives you back some deterministic example; a circular array is the simplest implementation of this interface (but there are others).
You could also do simple load balancing by sending work to Worker[workCount++]. While usually you want to track each workers' existing workload (because the work takes unpredictable time), this simple approach could be sufficient if all your work completes in about the same time.
If you're doing fancy math or science computing, you may be working with finite groups or fields, whose elements you could stick in an N-dimensional circular array (based on the characteristics of the field).