I always watch an anime multiple times, never back to back though. I give it time to sink in, watch other anime, then I'll come back with a fresh set of eyeballs, and I have a blast finding all the little tidbits I missed during the previous times watching.
In fact, now I have yet another reason to rewatch anime... my wiki. As a fan-fiction writer, having quick and easy access to little details can help add flair to any story.