When I was an undergrad, Jon Borwein showed me the inverse symbolic calculator. You type in a number, and the inverse calculator tells you what it 'thinks' your number is, for example $\pi^2$. I thought it was a useful tool for experimenting. For example, if you have a series or integral which you can't evaluate exactly, you can evaluate it numerically and plug it into the inverse calculator to see if it spits out a 'meaningful' answer. You then have something that you can actually try to prove.
This isn't a very deep application, but there is more information on this page for experimental mathematics.