Which brand of skepticism is the hardest to refute or argue against? What argument has given the other theories of knowledge the hardest time?
Which brand is hardest to refute is going to depend on what epistemological standpoint one is arguing from, but I imagine that Pyrrho and Sextus Empiricus represent two of the most enduring proponents of skepticism; certainly, anyone attempting to "refute" skepticism would have to grapple with Agrippa's Trilemma.
EDIT: Just to flesh out the link a bit--
Agrippa's Trilemma argues that all attempts at justification resolve to one of three cases: an infinite regression (e.g., we justify A by B, which is in turn justified by C, etc., with no end); a circular argument (e.g., we justify A by B, which is in turn justified by C, which is justified by A); or a set of unjustified axioms (we justify A by B, which is taken as axiomatic.)
In other words: if treated skeptically, nothing can be satisfactorily justified; all attempts at foundation are ultimately either assumed, circular, or deferred.