Top lane:
: He's sort of like a Master Yi who is more team fight oriented. Decent sustained damage and huge burst with a great ult for team fighting means he scales well and can come back into the game if he falls behind.
A bit harder to play effectively than most, but GP's late game is ridiculous. Huge armour-shredding AOE crits hurt like hell, and GP can easily kill a carry who isn't full HP with a single barrel.
Also applies mid. It's really easy to fail with Yasuo, but he can snowball hard and his late game damage and team fight utility are top notch.
She's not very good right now, but her late game is very powerful. Her huge AOE heal and ability to 100-0 tanks is not to be underestimated.
Also not very good, but his late game is still very powerful.
All right, so let me fix a misconception people have about Tryndamere: his late game isn't actually particularly powerful. It's not due to his damage. In terms of damage, Tryndamere does more damage per auto attack than any of these champions, is very hard to duel, and is a splitpush king. In terms of damage alone, he's on par with a hypercarry. But see, here's the thing: late game champions need to be able to team fight. Tryndamere can't team fight very well. He's too easy to kite compared to, say Yi. So even though Tryndamere can kill Yi 1v1, Yi is better late game.
Jungle: Late game carry junglers aren't good right now, to be honest, but of the late game carry (formerly devourer) junglers Yi probably had the best late game.
Mid:
Insane DPS to shred tanks, burst to kill squishies, mobility, a shield, self peel, hard CC, Azir has it all. His late game is one of the most obnoxious in the game.
No surprise here. Veigar infinitely scales like Nasus, but his overall scaling is even better since all of his damage abilities scale off of AP.
Karthus' damage is obnoxious obviously, his AOE DPS is insane, but honestly IMO his real value in team fights is his passive. It allows him to unload his DPS for I think five more seconds in a team fight if killed. This basically means that the enemy team is faced with the hard choice of focusing the late game magic damage DPS powerhouse and then being subject to his passive, or letting him run havoc on their team.
Probably the hardest scaling assassin due to his high burst, insane mobility, and low cooldowns.
ADC:
The best late game ADC IMHO. Vayne probably has higher single target DPS, but Twitch's assassination ability, huge range, and ability to auto attack an entire team (coupled with strong steroids) make him a team fight monster.
The best duelist among ADCs, and if properly peeled for can kill entire teams, one person at a time. Shreds tanks and squishies alike.
that will cut you out of farm or 
top but nobody plays him top these days and with right itemization nasus can do miracles.
