I was on a flight to Aberdeen last year when someone I recognised also boarded. I didn't get a chance then to speak to her but I knew she was either one of the doctors' receptionists or she worked in a local shop. When we arrived, I spotted her again and realised it was actually Camilla, Duchess of Cornwall! The previous year, prince Edward's wife and children were on the same flight as us.
Dd1 flew as an Unaccompanied Minor to New York when she was 12yo. She kept bumping into Sean Connery at the airport and on board. He was very nice to her!
We saw David Hasselhoff in a party goods store in LA and saw Arnold Swartzenegger in his Hummer on an LA freeway. That was a bit embarrassing as it was heavy traffic and we kept passing him, then he'd pass us and so on, and really, there are only so many times you can wave at someone. 
I saw Nick Knowles shopping in Windsor last year on 23rd Dec with his wife and child. He was laden down with bags - last minute panic buying!
My red letter day, though, has to be the one when I met Emma Thompson, Phyllida Law, Greg Wise and the swooningly gorgeous Alan Rickman. I'm still so sad about him, what a loss. 