One of the most awesome fixed-point theorems I know of is due to Pataraia:
- If $L$ is a poset with a bottom element and with joins of directed subsets, then every monotone function $f: L \to L$ has a (least) fixed point.
It is a strengthening of the Knaster-Tarski theorem, and is somewhat reminiscent of the Bourbaki-Witt theorem, but is entirely constructive. Related discussion at the n-Category Café here.
