I used to work in theater, so I've met a lot of people who are famous, or sort of famous, or famous in their own minds. Mostly, I either worked with them, or crewed a show they were in, or met them because they were backstage at a show I was working on.
My favorite "I met someone famous" story: I was taking notes at a dress rehearsal, for costume designer Marty Pakledinaz, at a show at the Public Theater in NY. After the show, he asked me to come back stage, to take notes on anything that actors needed. Starring in the show: Christopher Walken. Hanging out in his dressing room, waiting to go for drinks with him: Mikhail Baryshnikov. Marty introduced me as his assistant, and the guys asked of we wanted to go for drinks with them. Marty said no, and they guys both looked at me. I could barely squeak out an answer: no, it's kinda late, and I live in Brooklyn.