Amusement Park planet (or Shore Leave Planet) was the informal name given to a planet in the Omicron Delta region first visited by the USS Enterprise in early 2267. It was so named given its function, of drawing images from visitors' thoughts and manufacturing objects and characters by which they could act out their fantasies. Once this function was explained to Captain Kirk by the planet's Caretaker, the captain authorized shore leave for his crew. (TOS episode: "Shore Leave")
The Amusement Park planet became a popular destination for other Starfleet crews following this first visit, and the Enterprise crew revisited the planet on a regular basis as well.
Not long after this initial visit, the Caretaker died, and the planet's computer rebelled against its programming, becoming hostile toward visitors. Kirk and Spock were on hand to talk to it, and convinced it to fulfill its intended function. (TAS episode: "Once Upon a Planet")
During a subsequent visit, Enterprise Yeoman Minnie Moskowitz expressed a desire to remain on the Amusement Park planet, and the computer complied, creating a duplicate Moskowitz to return to shipboard duty. Moskowitz's psychiatrist, once learning of the switch, also considered giving up Starfleet and remaining on the planet, but ultimately returned to the ship. (TOS - Strange New Worlds I short story: "The Girl Who Controlled Gene Kelly's Feet")