A fail so epic it must be expressed in exponential terms. In this context, faillionth is not directly related to units of fail (failions), but is rather a very large fictitious number (e.g., bazillion, fafillion).
Man, did you actually make it to the end of GI JOE: The Rise of Cobra? The plot made absolutely no sense, the special effects sucked ass, and neither one of the hot chicks got naked the whole time. Fail to the faillionth power!
Did you see that Obama got awarded the Nobel Peace Prize? The Norwegians think they can influence global geopolitics by giving Obama their meaningless award, when reality is that nobody gives a damn about what Norway thinks because their economy is smaller than Hong Kong's and they spend less on their military than does Mali. Fail to the faillionth power!