I found burden of knowledge most interesting(I suppose one would expect you to get this one right first) and think Riot has done very poor job on that front.If you pay so much care not to burden people you should update data often and make sure it is accurate.This is one thing that amazes me as it is easy to do and you do it so poorly.Heroes knowledge base usually comes out week or so after hero,changes in heroes abilities are often not updated.Abilities don't have good descriptions(eg physical or magic damage - so many people ask about this again and again and still nothing is done about it).This is the most basic stuff that makes game much easier for new players and is easy to do(just time spent needed).Bad jobie you have done here indeed.
A good example of this is Twitch. If you've never played him before, he seems like Teemo. He's an auto-attacker with stealth who relies on movespeed bonuses and poison. He ambushes you and it's a one-on one fight, you start to destroy him, he's almost dead, and BOOM. 40% of your health is gone. Dead. The animation for Expunge is so small that it often gets lost in the particle chaos of a fight, and there's no effect when it's done to you either, except that the poison particles disappear, which is also very easy not to notice. It fits in well with his assassiny playstyle, but if you don't now how Twitch works, reading his ingame description doesn't allude to Expunge being anything like that instead of something like everyone else's highly visible moves.
Another one is Karthus's ult. Took me like forever until I knew that you could interrupt (though that's mostly because he pops it nowhere near me). Some of my RL friends considered a number of the other channeled moves to be some of the most overpowered things in the game until they realized that they can be interrupted.