Physorg has an article about two scientists who built a supercomputer using 8 Sony Playstation3 consoles. The instructions for creating your own are here, released under a Free/Open Source license.
So if we can find eight PS3's that no one is doing anything with, we're all set. The other route that we could go is to use older PS2's, which we could get cheap from gamers upgrading to newer/faster hardware. We could build a beowulf cluster using 70 PS2's like the National Center for Supercomputing Applications did a few years back.
Once we get it built, I'm thinking the first problem we can work on is to simulate the massive nuclear explosion at the earth's core that a recent theory postulates as the cause of the moon's formation. This satisfies two goals -- research on something that goes bang, and research on something that might further the Ultimate Goal of World Domination.