I have never actually met (been introduced to, shook hands, etc) anyone really famous that I can remember... but I have gotten close.
Before I tell that story, I wanna say these things: My friend at school recently had dinner with Robin Williams. My parents stood in line behind Vanna White at a movie theater in LA a long time ago. I was pretty close (distance, not personally) to Brett Favre when I went to the Denver/Green Bay football game about a month ago. My grandma has seen Tiger Woods up close at an airport. My parents saw Mike Myers in a Vegas casino once. A kid at my school's parents know the Bush family pretty well (yeah, George Bush, the president).
Now the big story: My marching band took a trip to Indiana last year, and we were in Denver International Airport flying out to Chicago. We were waiting at the gate, all 150+ of us, for our plane to dock and get cleaned so we could board. I was sitting with my back to the wall looking out at the other gates across the movable walkway.
All of a sudden a monster walked very swiftly across my field of view, helped along by the moving walkway. All I really had time to see were his black bandana covering his golden hair and the giant muscles on his arm. ...And his trademark shades and golden pointy 'stache.
And I said "Whoa, is that Hulk Hogan?"
My two friends decided to chase him down, no easy task for a man moving so fast. We weren't sure if it was actually him until they caught him and stopped him. They asked him if he was Hulk Hogan, and he confirmed the statement (apparently in a voice so low they could barely understand him) and they asked what he was doing in DIA. He muttered something unintelligible (thanks to the voice) then asked why they were there. They quickly said stuff about our marching band going to nationals and he said something along the lines of "oh cool, I love music" and handed them both 20 bucks out of his pocket and kept moving.
They said afterwards that they were scared he would squish them with his massive muscles and strength.
I was the one that saw him first. He was our idol for the rest of the trip.