If you walk around NYC, you see a celebrity almost every day. I used to work in a building at Columbus Circle until they tore it down for the Time Warner building. There was a production office on the same floor as mine. This was the building that had the car flip over in the second Die Hard movie (Die Harder?). I was in the elevator with Mel Gibson (he hid in the corner of this empty space) and I was at the urinal next to Ron Howard. I said, "how ya doin'," to Ron Howard. He responded with a, "hey," or something like that. I never have any interest in bothering these people.
One time I saw Martina Navratalova coming out of a bodega. I did a double take. She saw me look at her and shrunk back into the store. I kept walking. Sure, I took note, but I have no interest in doing more than that.
The most surprising person I ever saw was Frank Deford. Most people don't know him, he was an editor at Sports Illustrated and comments on sports on TV a lot, and I'm sure he's rarely ever stopped. I was walking on the Upper East Side of Manhattan and he was walking towards me. What struck me was how HUGE he was. He is 6' 4" and looks larger! Most actors you see are very short, usually surprisingly short.
Anyway, probably doesn't answer your question, but I usually just stay away.