In terms of just walking by someone: Robert De Niro. Just walked by him as he was sitting at a table in mid discussion with someone at a fancy Italian restaurant in Los Angeles.
Actual conversing with: Kirsten Dunst. This was way back in like '93 though. So I was around 13 and she was 12 or so. I had gotten dragged to some fund raiser for public education that was hosted at Anne Rice's house (author of Interview with the Vampire) due to my mother (who worked in the Dept. of Ed. in Louisiana) being invited. It was some really fancy black-tie event and Dunst and I were the only two "kids" there. So we just went in the other room and watched tv and bullshitted for three hours or something. I just remember asking her what it was like to be an actress (she were in New Orleans for the filming of Interview), and her being pretty cool. And I remember her being cool about me watching the Saint's game, which I'm sure wasn't her first choice.
Alternatively: George H.W. Bush. About 6 years ago I was clerking for a Federal Judge in Houston. Went to some private members only club for lunch (judge loved the place, and it if was a working lunch, it was always his call as to where we ate). Small place on like the 30th floor of some downtown Houston building, with only half a dozen tables. H.W. was at the next table one time. Him and the judge knew each other, so the judge introduced me. Was a quick 2-3 minute conversation with the usual "Where you from? What made you want to practice law?"
Most Random: I answered the phone when Bill Clinton was on the other line. This was during his presidency. I was at some downtown hotel in Baton Rouge. It was election night, and a buddy of mine's father had just lost a close race for the US Senate. He and Clinton had known each other due to them working in state government in Louisiana and neighboring Arkansas at the same time. So Clinton was calling to offer his condolences. He called the hotel suite and I was just the one closest to the phone, so I answered. He just asked, "Could you put John Smith on the phone?" And I remember thinking as I passed it over, "That guy really sounds like the President." Then my buddy's father starts talking with, "Yes Mr. President. Thank you Mr. President" etc.