Harry Potter and the Methods of Rationality is a fanfiction by "Less Wrong" (a pen name of Eliezer Yudkowsky) with a premise that the orphaned Harry Potter has been raised by a scientist stepfather and now has the equivalent rationalist skill of roughly an 18-year-old Eliezer or thereabouts, i.e., massively flawed but still pretty darned impressive. Naturally the hero's challenges have been stepped up in difficulty as well.
It would be fair to understate that this fic has proven itself popular.
TV Tropes page here (warning, potential spoilers).
(This page also serves as a placeholder for subpages which can be linked to from fanfiction.net, so as to provide supplementary information.)
Current thread (Ch 96), ch 95, ch 94, ch 93, ch 91–92, ch 90, ch 88–89, ch 87, ch 86, ch 85, ch 84, ch 82–83, ch 81, ch 80, ch 79, part 10, part 9, part 8, part 7, part 6, part 5, part 4, part 3, part 2, part 1