I focus more on Film and Dramas so i don't know too much about games I'm afraid.
It may be worth just looking through either the Play asia or Yesasia site at the games section and seeing if anything interests you.
I say to check those 2 sites because I don't think Amazon can ship games outside of Japan.
The first Japanese game I got was Zelda on the DS, because of its furigana popups when you touch the kanji.... That and I'm a massive Zelda fan anyway.
I think it's Japanese title was ゼルダの伝説 夢幻の砂時計
I found it really fun because you had to use information that you read on signs etc. to solve puzzles.
I was very happy when I solved my first puzzle in Japanese
There are also games like Pokemon which only really use Kana since they are meant for children.
And there are also Japanese versions of games like Oblivion and Fallout 3 although I
wouldn't have though you would get any help with the Kanji in those games.
I personally like games like Metal Gear, but i don't think you can really pause any text in those type of games
Having said that don't worry too much about your ability to read everything in a game.
If it interests you then it's worth a try, there's nothing worse in my opinion than avoiding things just because of your "Level"