The general consensus is that City is the best Arkham game, and I do agree, but I don't think there was a perfect one because each one of them outshone the others in particular ways:

If I had to choose a favorite, it actually would be Asylum, even with all its defficiencies: Fighting mechanics were a fucking mess, and in reality there are just two bosses: Poison Ivy and everyone else mimicking Bane. However it's very different than the rest of the series, i.e. it leans more toward horror instead of action-adventure. The whole experience was claustrophobic and terrifying, a visceral feeling I never got from any of the others.

Origins is definitely the poorest-rated game, and for good reason. It comes after City, yet the graphics are terrible and simple things like grappling don't work right. However it had some of the best cut scenes of the entire series: discovering the identity of "Black Mask," Electrocutioner being thrown out the window, Joker talking to Harley after Batman saves him... And the boss fights were so amazing and plentiful: Deathstroke as an insane introduction, Firefly on the bridge, and needing to sneak around Bane with a time limit. I can't just dismiss it.

Honestly I would've said Knight was the best of the series if not for that damned Batmobile. Too much emphasis on that. Otherwise, really great to see all of Gotham, and it looks A-MA-ZING, with giant statues on bridges, beautiful gothic architecture, and very different feel on each of the islands. Although it didn't have as many dungeons as the other games, it had probably the best one in the series: the movie studio, where Batman reminisces about Jason, has a teamup with Tim, and having Joker serenade him.