Harry Porter built a functional computer out of relays -- including memory, ALU, registers, and all.

That's quite a project to undertake. It also gives one a solid understanding of just how the original electronic calculation devices were made and how complicated they were. Some stats from this project:

• 415 Relays
• 111 Switches
• 350 LEDs

That's alot of wiring, but the nice thing is, that once it's working, there is probably very little to go wrong which cannot be seen or heard, so that tangle of wires should never need to be debugged (which is fortunate with the wires being mostly the same color!)