Ignoring that these exercises are generally embarrassing and uncomfortable, the "best" one I had was in a moderately sized group of students (20-25) where on the first day everyone had to find an alliteration with their first name initial and a food they liked, and introduce themselves with it. (Example: Fred + frittatas).

Virtually everyone has a food they like, yet it’s not an overly personal question, and the little alliteration helps immensely in memorizing new names and faces. I knew almost all names from this by day 2.