I once asked my honours real analysis class to define the concept of an integer to a hypothetical bright young kid who was already perfectly familiar with the natural numbers and the operations one could perform on them, but had not yet been exposed to negative numbers. The response was both enthusiastic and chaotic; I remember one student, for instance, giving a heuristic to explain why the product of two negative numbers was positive, which was interesting but not directly useful for the problem at hand.
Nevertheless, the question served its purpose; when I did then introduce a rigorous definition of the integers (as formal differences of natural numbers, quotiented by equivalence), the need for such a formal definition was made much clearer by the lack of an "obvious" way to do it by other means. And I think it also had a residual effect in motivating the fancier epsilon-delta definitions that arose later in the course.
Another example I have seen, at the early high school level, is to challenge students to come up with a watertight definition of a rectangle. This is remarkably difficult to do for students without training in higher mathematics; not only does one have to deal with degenerate cases (e.g. line segments), but often crucial properties (e.g. that the four sides of a rectangle have to be connected at the vertices) are omitted. One can also get into interesting debates, such as whether a square should be considered a rectangle.