I'd love to see a replay feature added. I was actually extremely surprised to not find it available when I started playing.
I've read reference to a 3rd-party program that can do this but I won't use such a thing for fear of keyloggers/worse and potential ToU violations.
I realize that permanent remote storage is expensive, but for replays I can't see any reason why players can't store them locally.
The server must store replay data as a game progresses or you wouldn't be able to spectate with a delay.. just give us the option to store it locally after a game before it's purged from the server.
This would be my number one focus for the devs for 2013. Over maps, balance, champs, skins, ...
Must have replays, please.