What defines a hyper carry?

In League of Legendes, what defines a hyper carry?
What the champions can fit the role and why it isn't used more frequently?

A "Hyper Carry" by definition is someone that scales so well into Late Game, that he becomes unstoppable.

In LoL, there are no True Hyper Carries, due to the fact that all carries can be focused and shut down at some point, although if we look at Vayne, she can be defined as a "Hard Carry" (amongst others) due to the fact that she reaches a higher peak at end game (with her True Damage, which scales with ATK speed, which you can find on the common ADC items such as SotD and Phantom Dancer)

There are two exceptions, mainly due to the fact that a fed champion "Can be shut down by CC". This does not apply to Irelia and Olaf. Once one of these has an advantage in gold and levels (aka "fed"), they are very VERY hard to stop, mainly due to their damage output, tankiness, sustain, and immunity to CC.

So, conclusion:

• LoL has "No Hyper Carries" (Faceless Void waves Hello)
• LoL has several champions with a higher potential when fed. (Olaf, Irelia for example)
• LoL also has simple carries that naturally reach a higher peak at end-game than other, known as "Hard Carries" or "Late Game Carries". (Vayne, Kog'maw and others)

Additional details:

Due to popular demand, I am now apparently forced to list the most dangerous champions to feed.

1. Irelia - Heavy CC reduction, innate tankiness and regeneration, True damage. Very hard to stop after she has an advantage.
2. Olaf - Short CC Immunity + tankiness buff, heavy True Damage on demand, very VERY tough once at an advantage (the less Current HP he has, the faster he attacks. The more MAX HP he has, the more damage he deals)
3. Farm-carries: Nasus, Sion and Veigar - Long story short, they have good CC and an INFINITE stat potential due to the fact that their stats scale with their farm (Nasus Q Bonus Damage, Sion HP (with Atma's, AD as well), Veigar Ability Power)

Gimme moar and I'll fill the list

Hyper-carry is actually an old/obsolete concept that no longer applies.

Basically, regardless of how well/poorly a champion was doing, after hitting the 50-55 minute mark, certain champions become incredibly strong.

This was prevalent during season 1/early or pre-season 2. Champions like Olaf, Irelia and especially) Master Yi could have catastrophic impact in teamfights/duels once they finish the bulk of their build (for Irelia that meant Trinity+Survivability; for Yi it meant Attack Speed + Life steal; and I never played Olaf back then so I can't comment) and regardless of how hard they had been shut down throughout the game, they will tear you apart after that point.

Basically what I'm trying to say is that Irelia could jump in vs 5, kill 2 and be on the verge of killing the 3rd before being shut down. And when I say jump in vs 5, I mean SOLO vs 5; so you can imagine.

Item and champion balancing has taken care of that though.

Bit of nostalgia: at the time when i was playing Irelia, the phrase 'better nerf Irelia' was already lol-lingo so you can imagine how incredible she was.

Well, I don't know about "hyper", but if it's similar to hard carry - carry, who can carry hardest with full inventory and max lvl, I think it would be Tristana. She can easily reach max as with less as items, her range max lvl is awesome and she has escape. You just stand behind your team and kill enemies with max as.