Here's how I would handle this, but it requires at least four players. You also won't be able to do this with more players than the maximum of any game (i.e.: if the maximum player limit on your version of Catan is 4, you can't do this with five players).
What you'll need: a 60-second sand timer and all three games.
You're essentially going to be playing parallel games, rotating turn order in the same direction on each board. The fourth (and fifth and sixth, if available) player, who is not the main player on any board, will be the referee to make sure there's no funny business and to make die rolls on the Risk board. Turn order would look something like this:
- P1 on Catan; P2 on Monopoly; P3 on Risk; P4 ref
- P2 on Catan; P3 on Monopoly; P4 on Risk; P1 ref
- P3 on Catan; P4 on Monopoly; P1 on Risk; P2 ref
- P4 on Catan; P1 on Monopoly; P2 on Risk; P3 ref
- Back to step 1
Each player takes their turn on their main board. Risk turns generally take the longest, so that's why the ref moves to Risk; they'll already have a sense of the board state. When necessary, the ref will also roll dice against the main players, which will mostly be happening in Risk.
If an action by a main player requires a main player from another board, grab the sand timer. The players have 60 seconds to finish that task, whether it's trading in Catan, or payment in Monopoly. During these 60-second interruptions, all play should be suspended on any board where the action isn't taking place. At the end of 60 seconds, if the action isn't completed (e.g.: a trade isn't agreed upon), the action is cancelled and can't be repeated again that turn.
At the end of the game, you have up to three winners, one for each of the boards. You also have no friends, but that's a different issue.
Note that this technique could technically be implemented with other games where:
- Player interaction is fairly limited
- People take complete turns before the next player
- Turn order doesn't vary between rounds
Wanna throw Ticket to Ride in instead of Monopoly? That'll work. Not a big fan of Catan? Try out Candy Land instead.