No, what I'm thinking of are those unscripted moments that come up organically, moments when the game is just trying to be a game, and only by accident does it convey something more. These are obviously going to be highly personal and subjective; you'll have played the same games and have similar stories, but all of our little moments are going to be unique. Here's my top five.
Rocket Geometry in the Mine Tunnels of Half-Life
Half-Life is a superb first-person shooter filled with great moments, both scripted and organic. Most of the organic ones are based on the extremely good level design, which provides both a great sense of place and an incredible environment for shooting baddies.
Except for the mine tunnels. The mine tunnels sucked.
Long, tedious, and painful, the video game equivalent of a root canal being performed rectally.