I had a couple of encounters with Paul in NYC, but I'm afraid they're not the most pleasant... Here's the first one:
In February of 1993, Paul was doing SATURDAY NIGHT LIVE. I found out the hotel he was staying at, and so my wife and I hung out with about 20 other fans hoping to get an autograph.
Because I was such a lifelong Beatles fan, I stood in the February cold for two days (not consecutively, of course - I went home in between) and what Paul did was play "cat and mouse" with us all day. He would leave the hotel to get in his limo and go for a jog, then return to the hotel. We'd wait three hours, he'd come back out to go to lunch, then come back. Another couple of hours we wait -- he goes out to dinner. He comes back to the hotel, stays a while, goes back out to rehearse at NBC.
To make a long story short, during the course of the two days there'd be as many as 20 people there at some times and as little as 3 at others. Each time, Paul would say "hello!" or "good morning!" but then he'd be off, without stopping.
One time, I jokingly yelled to him (lying): "Please, Paul - I came all the way from Canada!!" -- and as McCartney climbed into the limo, he turned to me and shrugged: "Oh, did ya? I came all the way from England!" - and then the limo sped off!
As it turned out, I actually DID get Paul's authograph that weekend -- but it was only while he was inside the lobby, and he sent his PR man John Hammil outside to collect items for signing. Paul signed them INSIDE the hotel, then the items were returned to us. As Paul left for SATURDAY NIGHT LIVE, he yelled to the crowd: "okay! Everyone got one?!" - and we all said "thank you's". It wasn't the best way to get his autograph, but it was better than nothing!