What champions winrate increases the most with games played?
I would guess it will be a complex champion like azir or one with a hard positioning like Cassiopeia, but I may be completely wrong.
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Look at champion.gg
Possibly Master Yi, but I'm not going to look through all the champions to verify :D
I'd bet 
is definitely up there http://champion.gg/champion/Rengar
http://champion.gg/champion/Kassadin Is another contender.
http://champion.gg/champion/RekSai is another.
Lol here you go, http://champion.gg/champion/Karthus/Top Karthus top goes from 38% to almost 71%
Lee Sin.
Probably someone like Master Yi if you look at how much more win % you have when going from the 1st time at a champion to the 2nd. But if you're looking at going from the 100th time to the 101st time, then probably someone more like Lee Sin. I don't feel like there's simply one champion that always has the greatest marginal change in win rate.
If we control for a decent playrate, let's say greater than 5%, then the answers are Rengar jungle, scaling from 46% to 59%, Yasuo mid with 38.5% to 50%, and Fizz mid with 46% to 57%, Riven top with 41% to 51.5%, Master Yi jungle with 47.5% to 59%, Jax top with 45% to 58.5%, Nidalee jungle with 38% to 51%, Kindred jungle with 46% to 67% (Kindred OP AF), and Viktor mid 41% to 55.5%. Notice how the majority of those champs are considered to be faceroll easy.
with veigar, if you look at plat+ players right now, when you go from 0 plays (42% win rate) to 125+ plays you too could average a 58% win rate with him. isn't that great!?
http://champion.gg/champion/veigar
(despite his 40% win rate in 0-25minute games, and only 51% in 40+ minute games)
Ryze Scales from 35% to 76% which is pretty busted. It might not be the new ryze since it says (2014-2015 Ranked Experience)__ . Does someone have a link to the new ryze ?
Ryze and Gnar are both very high up!