I used to live in New York City so you see random people trying to keep a low profile on the street all the time, but my favorite story belongs to one of my friends who tends to be pretty clueless about pop culture. She went to a neighboring friend's party and someone at the party ended up crashing on the couch, and all three of them went to brunch the next day. While they were at lunch people kept coming up to the window and taking pictures, so finally she went off on how she knows the restaurant was renovated recently but it's not THAT nice, why do so many people want to take pictures of it? Turns out the dude who crashed on the couch was Jake Gyllenhaal.
Edit: Since people seem to enjoy this story I'll add in another one. This is all highly ironic because my friend worked as a drummer in the house band for a talk show in this same era. It was a running joke on production that she never had any idea who the guests were, so they would have her give all the guests nicknames. I remember her trying to describe the guests on one of the shows to me because she still didn't know who they were, but she had named one of them "Muscles" and one of the "Cap", because one of them was strong and one was wearing a hat (I didn't say these were good names), and Cap was a bit perturbed that the other guy got a nickname like "Muscles" and he was simply relegated to "Cap". It turned out she was talking about Chris Hemsworth and Zac Efron.