Great game! Love the simplicity!
I played a round, and got to 512. But toward the end I wasn't sure if I was actually playing with a strategy, or just pressing buttons randomly with some thinking involved.
So I built a script to randomly press the arrow keys[0]! I let it play a few games, and the highest it got to was 128 before consistently losing. So I guess you'll need some decent strategy to get to 2048.