My favorite game of all time would have to be GoldenEye 007 for the N64. But that was released in 1997, so it probably won't be considered "classic."
In that case, my favorite classic game is Jungle Strike. You get to blow up the White House. It's just awesome.
As for the arcade, I'm going to go with the first game I ever played, Asteroids. I was stuck with the "Asteroids Deluxe" one for a while (it sucked compared to the original), but once I saw the original I did a :o and loved it.