Only Pocket has async (grrrr @ TTR not being universal). It's GC-based with everything that goes along with that (notifications disappear once you've seen them). Pretty good async game in general though and the GAF community is decently active. In terms of current board-game activity on GAF I'd put it at:
1. Carcassonne
2. Ascension
3. Nightfall (new hotness factor)
4. TTR Pocket
5. Tigris & Euphrates (just b/c I've been bugging people to play it)
But it's super easy to drop in to the BG thread and get games started, or just send out invites to people on the spreadsheet, especially for a popular game like TTR Pocket. Game itself you can probably learn in about 10m.
Maybe I'm misunderstanding the term asynchronous - I tout it meant you can play the game without being live with other players, like you take a turn and leave it, while others play.
ascension allows that doesn't it? And carcasonne, I thought.
Edit: oh you're talking about TTR versions. Disregard!