Batman: Arkham Origins
Absolutely amazing series!
Though I did not ever play Arkham Asylum myself, I did watch an entire series on it and an additional series on Arkham City before actually playing Arkham City, for PC at least, not sure if I tried it on Xbox 360 before the series ended. Anyway, the series tells an amazing Batman story and features probably one of the best depiction of the true meaning of Batman of all time. I also highly recommend getting the game on PC, since you can change Batman skins, in Arkham City at least, which in my opinion, can help keep the game more interesting for a little while longer.