Hello World, Need some good Anime.

Quick search yielded this.

I can also recommend you some good forum searches :P

clicky 1
clicky 2

http://anidb.net - LOTS of search options there ;)

Steins;Gate.After anime watch the movie "Steins;Gate - Fuka Ryouiki no Déjà vu".

It's an anime that you'll remember forever.

Devil May Cry anime was pretty good from what I saw. Doesn't have much to do with the orignal games though.
Also from same studio (Madhouse, who overall are pretty awesome), not game related but highly recommended anyway: Monster

Game related?
.hack//Sign
Log Horizon

If u want a browser based one to kill time in between.. try this One Piece Game

If you are open to something else than entirely game related anime, you should definitely watch Hunter X Hunter. About 40~50 episodes in there is actually a whole arc that takes place inside a game (Greed Island).

Awesome anime and there are tons of episodes + it's still airing ;)

The Persona 4 Anime is quite good but I fear you have to play the game before to share this opinion ^^

Space Dandy and Kill La Kill are my favorite series right know.

Btoom, gundam uc - spin offs, madoka magica xD

You should go to "HelpmeIwantanimebutican'tusegoogle.com".