I agree with André Henriques regarding algebraic topology. In addition to cohomology of $K(\mathbb{Z},n)$ data, I'd also like to see pages of spectral sequences computing various other things of interest, e.g. homotopy groups of spheres, complex cobordism, modules over real $K$-theory, etc. This subject has a massive dearth of examples, especially of examples using spectral sequences. One place where some computations exist is Bob Brunor's Bruner's webpage, and I like the way he displays some of those spectral sequences as JPEGs. I envision something like that going into this database, but with more computations because more mathematicians can contribute.
In a related vein, I would LOVE to have a repository of applets and other computer code mathematicians have written to help them do computation. For instance, Aaron Mazel-Gee has an Adem relation calculator which would have helped me a ton back when I was starting to learn this material. I've found various other applets (e.g. from Christian Nassau's webpage linked from Bob Brunor's Bruner's page) but it's not easy. If we had a database it seems it would be a natural place to put up code we've written along with documentation to help others with similar computations. Obviously algebraists have SAGE and GAP but there seems to be nothing like that for algebraic topology. If there is such a place please let me know, especially if it helps with the bookkeeping in spectral sequence computations!
