I’m thinking about what my favorite small games are — games that are simple and small in components (like Go) but also that offer me something without demanding every shred of my attention (unlike Go).
Hive is one. An elegant little puzzle box of hexagonal bug tiles trying to surround the other player’s queen. Coup is another: a bushido-like game of bluff. One wrong move and you’re watching the rest of the (short) round.
For cooperative games, the Crew (original) is brilliant: all the delicious tension of a partner trick taking game except everyone is on the same team. You’ll be thinking as loud as you can without giving up your poker face.