In Lua and Python it's a very natural thing to do when "just coding", because the moment you reference something that's not a parameter, you're making a closure. (so most of these will be quite dull as examples.)
As for a concrete case, imagine an undo/redo system, where the steps are pairs of (undo(), redo()) closures. The more cumbersome ways of doing that might be to either: (a) Make unredoable classes have a special method with universally dorky arguments, or (b) subclass UnReDoOperation umpteen times.
Another concrete example is infinite lists: Instead of working with genericized containers, you frob a function that retrieves the next element. (this is part of the power of iterators.) In this case you can either keep just little bit of state (the next integer, for the list-of-all-nonnegative-integers or similar) or a reference to a position in an actual container. Either way, it's a function that references something that is outside itself. (in the infinite-list case, the state variables must be closure variables, because otherwise they'd be clean for every call)
