I gave my Calculus II class a problem which introduced the Koch Snowflake and asked them to compute the area. They enjoyed it, and I think that had something to do with seeing an actual, geometric application of the infinite geometric series formula. Also: I loved the expression on their faces when I told them it had infinite perimeter.
I also gave them Gabriel's Horn as an in-class group exercise on surfaces of revolution and limits in general. This was earlier in the semester than the Koch problem, but it didn't have the same pizazz, even when I phrased it as: ``you can fill it with paint, but you can't paint it.'' I think they secretly believed limits were simply evil and that they should not trust things proven with limits.