When I read the title of this thread I thought it said "Hipster Carries". I'm kind of disappointed now that it's not about how Graves drinks Pabst and b*tches about obscure bands.

Cho'gath, have around 600-700 AP, 150 of both resistances, and 3.6k HP, proceed to poke with rupture and kill anything that comes within five feet

to 3vs1+ you basically need a fed bruiser. actually, when i get fed with nasus i can easily 2vs1 and sometimes 3vs1, but it all depends on the ennemy team.

The old jax. with a DODGE centric kit.

I'd see him in game then proceed to cry in terror as he beat the **** out everyone while dodging all of our attacks as well.

Thank god dodge is gone...

For the longest time I've been saying Skarner is one of the best 3v1 or more champs in the game. Like one poster before me said, having short CD's is a really important part of being able to fight 3 or more people. Seeing as how Skarner shortens the length of ALL of his CD's on auto attacks, it allows for his important skills to come up VERY fast (Read: Shield).

i think.. something like a super fed bruiser irelia can go 1 vs 3.. well it depends on the enemies too.. 1 vs 3 still can be done even in competitive play i guess.. (the enemy pro but it just like a non fed adc,supp and a rather squishy jungler) but 1 vs 5 is just like impossible in competitive play.. but well it possible in the blind pick with bunch of noobs and you like a super fed rengar/jax/xin that just jump and kill anything

Um karthus can 5v1 and win.... He has 2 cd's, his wall and his ult. His e is a toggle that deals tons of damage and his q has a 1 second cd that deals 400-800 damage.

You can just about "carry" with anyone depending on how good you are.

These are just the preferred characters all squishy and just CC or just OP.
-Riven - I don't really need to get into details. About the only major champ that can truly counter her is Olaf and Renekton. However, even against Olaf, Riven can do well as long as you can dodge his axes and time your Q, W, E well. Also, with her ult and the right masteries/runes, it'll be VERY easy for you to kill multiple enemy champions during a team fight; even when you have like 5-10% hp left.

-Fizz as said before. Fizz is amazing and I carried a game with him. However, it will be VERY easy to become arrogant and over-commit. Just do not do that and use your ult very wisely. Have your teammates hide in good location and you "initiate" to bait them to attack you. Back up a bit and find a good position to ult from. Ping for them to attack right before you shoot your ult where the cluster of enemies are. Let your teammates pwn them for a second or two, then charge in with your Q, use your W accordingly and you can even use Playful Trickster if you don't need to escape and prefer to use that AoE damage.

-JAX - I don't even need to describe why you can carry games with him. Just learn to use his skills well and you won't have a problem. Jax is a very amazing top laner.

-Irelia - Irelia IMO is one of the better champs to 1v3 or sometimes 1v4.

-Diana - easy to shut down but a good Diana player will just PWN everyone. I'm not that great but I have had games where I could shut down the entire enemy team myself with good kiting and usage of my Q, W and ult. I usually don't like telling peopel squishy champs can carry because a lot of people WILL overextend. Squishy champs like APC's can carry games hard too but it won't work if you are careless with them as they can still die easily. Good news about Diana is that she is generally tankier than most APC's because of her W spell. Fed Ezreal is easy to kill but it's hard to even touch a great Ez.

Jarvan can take on a team for awhile by himself if his build is really tanky.

Jax, Vayne, Trist, any of those absurd rangers, once you get your AS up, and get a decent amount of lifesteal they can take out entire teams. I can't think of any APs that really could, since most AP casters rely on their burst, after that, they are done until their CDs come down and usually thats plenty of time to waste down a squishy AP.