There are many, but this one came to mind first:
I took a road trip with a buddy to Houston, to watch JP Losman and Lee Evans put on a show on the field, to secure a win against the Texans. After the game, I went down close to the field and joined a group of Bills fans cheering on our victorious visiting team. I struck up a conversation with a woman next to me, and when the crowd started to die down, she and her husband started walking away, and she told me to follow them.
It turned out that she was Melvin Fowler's mother, and she brought me down to the family section, where the team was coming out of the stadium to get on the bus. She introduced me to a bunch of players and their families, I got some autographs on my hat, and had a really cool experience in general.
I have since been fortunate enough to do this a couple of more times, through other connections, but that first time was so random, that it most memorable.