I think that this (the MO question) is potentially a good question, or, in other words, it can be made into a good question. It should be made, a little more focused and then I don't think that it will represent a huge list of answers. Then it can be quite a useful and educating list. (Also the proof-theory tag does not seem to belong there. Update: it was removed)
The Calabi-Yau example was not so convincing (and so are the examples from physics) and it could be better to have a small list of good motivating examples.