Answer:
Gabriel's horn
Step-by-step explanation:
What the question is asking for is a 3-dimensional shape that has a lot of surface area and only a little volume. The first thing I thought of was a geometric figure called Gabriel's Horn. A quick search on Wikipedia will explain better than I can, but basically it has infinite surface area but finite volume. You probably couldn't test the shape in any experiment, though, since it's only theoretical and can't exist in real life.
Let me know if I can explain anything in a clearer way!