Monkey-in-the-middle was a type of game where a group of players tried to keep the ball away from one of their number.
In 2152, when asked by Travis Mayweather, who was trying to explain the Dekendi sport he got injured while playing to Dr. Phlox, the doctor was not sure if Denobulans played monkey-in-the-middle. (ENT: "Stigma")
The Star Trek Encyclopedia (4th ed., vol. 2, p. 17) identitifed this children's game from Earth as "man-in-the-middle". In this game, "a group of players form a circle around a single player. The players in the circle pass a ball between themselves, while the player in the center attempts to intercept it."