I haven’t tried to narrow this down perfectly (partly because I don’t think I could), but I’m going to say:
- Hive
- Carcassonne
- Hansa Teutonica
I can teach them to anyone. They cover a range of gameplay styles, durations, and player counts. They are all amongst my all-time favourite games. They each have a bunch of expansions (one of them on a fairly ridiculous scale). A limit of three games is pretty brutal, but if these were what I had at my disposal and someone wanted to play something, I think I’d be in good shape.