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Invisible skillshots: This one is straight up exploitable on Anivia, but exists on other champs and sucks for both the skillshotter and the dodger.
Karthus Suicide: He kills himself (somehow), robs the other team of Gold/Exp. Exploitable?
Messed up Recipes on DC: This one's TERRIBLE, and has been around since beta! You can't buy certain items once this happens, unless you keep DCing and luck it back to normal, which also makes you AFK a lot.
Messed up Minions on DC: Messes up map awareness, you have to ignore the "fake" minions, can really throw you off in certain moments. Since Beta.
Anivas wall-trap: Ends up being the longest root in the game, usually instant death for whoever it happens to. I think it's exploitable, but hard to pull off. Since Beta.
Bad Ping "hacks": Champion flickers around the map, practically impossible to kill/chase since he's not really where he appears to be.
Lemme know if there's more.
If I could choose just one to fix, it would be the items on disconnect. This one is seriously smash-your-keyboard frusterating!