Prince
Prince was pretty quiet about his needs for the majority of his career. But, back in 2007, he requested that the venue build him a "luxury five-bedroom home nearby" (this was denied). In 2004, he requested that “all items in the dressing room must be covered by clear plastic wrap until uncovered by the main artist."
Cher
Cher refuses to wear a backstage pass (she also doesn't let her crew wear backstage passes either for some unknown reason). Instead, she requests, in her rider, that the employees of the venue familiarize themselves with the faces and names of the crewmembers that she brings along. That, and she requests a separate dressing room just for her wigs.