OK, one crazy one was a Datum mag tape controller on a PDP-11 computer. These had one big standard board and a smaller plug-in board to customize it to the specific computer. The logic was just a whole bunch of TTL random logic, no microcode. It was totally dead, all registers read back as zero when you wrote to them. So, I started looking at the global reset. It was not being reset by the CPU, but the buffered side on the board was held in reset (low). I changed a few chips, and that didn't fix it. So, I then measured resistance to ground with power off, and it was REALLY low, like 2 Ohms. I started cutting traces, and finally got down to a 2" stretch of PCB trace that was shorted to ground. There was NO grounded trace within 1/4" of that trace that was at ground. I peeled the trace off the board, there was no speck of copper visible anywhere. I finally put a piece of wire wrap wire across where I'd peeled off the trace, being sure the insulation was intact. The unit worked fine until the end of its life.
That was one of the stranger ones.
Jon