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Coup you could just deal the new person 2 cards and start them with 3 cash, or just wait the few mins until the round is over and include them in the next one. The rounds are short enough that no-one will be waiting long to join.

I would suggest Mansions of Madness. With one change to the rules. The rules currently have the keeper draw threat for the number of investigators that started the game. This would have to be modified to be the number of players currently playing the game instead of investigators. This way the threat accumulation would be accurate to the amount of players. ...

Fluxx and a few other social card games allow players to join or leave at any time.

If you play Gipf with potentials then each time you activate a potential you play the game the potential if based on. If you win you are allowed to use the potential power. Note that these rules are optional (for one you would need to own the game, and it also takes a long time if you play with many potential). Also, while each game is played on its own all ...

Mansions of Madness has a number of 'puzzles' that are like mini-games. You have to solve them to uncover clues and gather other important things.

Trajan has you playing a round of Mancala each turn to decide what action you take, with bonuses for accumulating the correct colored pieces in certain locations.

The Shahrazad card in Magic the Gathering has the players stop playing the game and play another game of Magic with their remaining cards.