Surprised no one has said a ring game of 9 ball
BmoreMoney mentioned it. If the three players are pretty good, then the ring game is my favorite.
For a party to get people to socialize around the table, cut throat is great. I have played using all of the rules variations suggested above (not at the same time). The additional rules we followed are that the order in which players are knocked out determines their group in the next game (first out becomes 11-15, second out becomes 6-10, and winner breaks and gets 1-5). Also, we played that you can sink balls from your group and keep playing. That even allows you to sink the last ball in your group as long as you run out.
Golf is also a good game for 3 or more players.
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