In 1994, I was an undergraduate student in Perth, Western Australia studying linguistics and planning to continue on to do a PhD.
I found out that Noam Chomsky was coming to Sydney in early 1995 to give a number of lectures and, as I had relatives there, decided I'd book a flight and spend a few days visting them and attending the Chomsky lectures.
There was scant information about the exact dates online so I emailed Chomsky who kindly replied with the details of his talk schedule. We continued a short correspondence about possible PhD studies at MIT.
Come January, I flew across the country to Sydney. Chomsky's main linguistics lecture was at the University of New South Wales but when I got there, the lecture hall was so packed, they were putting people in overflow rooms with a video link.
I'd come all this way to hear the man speak and I watched him via a video link a building or two away!
I left the overflow room after the talk and was walking past the main lecture hall when Chomsky came out, swamped by hundreds of people that wanted to talk to him. I really wanted to meet him in person myself but I knew there was little chance, so while the crowd of people tried to get their thirty second chance to talk to him, I stood back and got chatting to one of the organizers of his visit.
I told the organizer I'd come from Perth and I was considering doing a PhD at MIT. "You should join us for morning tea!" she said.
Shortly after that, the crowds were told Chomsky had to leave and Chomsky, the organizers and me went off for a coffee and a muffin. I got to talk to him for almost half an hour about linguistics, a professor of mine who'd been his student and the possibilities of doing a PhD with him.
Oh, and Chomsky didn't eat his muffin, so I got a second one :-)