The first computer I built was a Southwest Technical Products 6800, back when you had to build your own memory cards and the big new thing was 4KB on a single card. Used it partly for making crappy electronic music but one of our friends was into video special effects so we ended up putting together a computer controlled video processing studio.
Second computer we built used a TI9900 (if I remember correctly), a 16-bit processor with hardware multiply and divide... that more or less became the geometry engine for the third system, which was an array of 24 6502's each doing rendering for a portion of the screen. We actually got it working, although it was a painful lesson in how reliability goes down as you add more components to a system. It was probably not too different from keeping an old vacuum tube computer running, although our problem was connectors rather than tubes.
I ran across a couple of YouTube videos recently showing off relay computers that people had built, and I have a growing urge to build one of those. Downside is that clock speed is on the order of a few Hz, and the voices in my head are saying that it would be a lot easier to just write a simulator instead of actually building one, but what can you do ?