Many people know Free Rice as a vocabulary-building program with a humanitarian bent: every time you get a question right, you donate 20 grains of rice to the World Food Programme. Now, however, Free Rice covers a host of subjects: grammar, math, geography, and, my most recent discovery, fine art.
You are given a famous painting and are asked (via a multiple-choice question) to identify the painter. Now, this may not be much of a game, per se, but I had to do a similar sort of assignment for a high school humanities class, and memorizing a huge number of famous painting and painters has proved remarkably useful to me.