Unlike DotA,
League of Legends is a team-based game.
It's possible to snow-ball so far ahead that no one can stop you, but the game's mechanics prevents you from winning a 1 v 5.

Your best bet towards victory is being able to communicate and work effectively as a team.

I've never seen any player carry any game. It's too difficult. I've seen players do really well and change the course of the game, but carry? No.
Carrying implies that you are completely changing the outcome of the match. You are the sole reason your team will succeed and every other member on your team is contributing little to nothing to the point where if you weren't there they would be stomped. And that doesn't happen. Because rarely is someone so good that they can make up for the mistakes of four other people as well as still come out positive against the opposing team.

Carrying, truly carrying, is almost nonexistent. There's a difference between doing reallly well and carrying.

Man I came in here hoping to find some great secret to winning. Oh well, was worth a try.