Hmm... I'd say probably Lee Sin is the best hyper-carry? A good lee sin is both melee and ranged, he's a healer/support/tank/carry - he does everything well, and has no weaknesses - and scales very well off gear. Apart from people dodging his feet/kicks, he's tanky enough to dive right into the middle of an enemy team and not care, and he's hurty enough to score kills by doing so.
As everyone mentioned, no champion is strong enough to do everything at the same time - but a decked out Lee Sin, Fiddlesticks, or Darius each have a combination of tankiness and hurtyness that would allow them to hypercarry, if super geared. Oh, and assassins can generally be super dangerous if they snowball - Akali, Nocturne, Evelynn - if you feed them kills and it gets out of hand.. they won't ever take on 1v5, but they'll pick of one or two of you over and over and over until your team clumps (and in doing so, loses lanes).