Dito Pataraia once came up with an elegant proof of (essentially) the Bourbaki-Witt theorem. The proof is very short, and deserves a look. Furthermore, it uses domain theory in an essential way: the core is showing that the set of inflationary monotonic maps on a dcpo is itself a dcpo, and is directed, hence has a supremum. Dito Pataraia would very rarely publish his findings, and we owe it to Martín Escardó that it did not disappear with him. I personally became aware of this thanks to one anonymous referee of the book. See the full post.