My favorite moment from another character in my RPGs this year was the previously mentioned monkey-seeking character in our Paranoia game.
Early in the game, the representatives of Alpha Complex sought to bribe a character into confessing his treachery. He said that it would depend on the price. He wanted a pet monkey. A good part of the early game was spent with the PC asking everyone if she or he was the person from whom he could get his promised monkey.
The GM smartly added an NPC with a monkey companion to escalate the situation. Through the middle part of the session, the potential monkey provided some incentive for the PC (in a childish, insane way that fit perfectly in the game). It even held up through the end game after the character recounted all he had done and now, NOW he wants his promised monkey.
That little bit of improvised dialogue became a major axis around which the humor and role-playing for the session rotated.
I can not think of a better example of how important improvisation is to the fun of RPGs.
This also got me thinking about the format for sessions. The Paranoia session was in-person and that seemed to affect my enjoyment. I have had fun with online sessions and plan more in the future. However, there was something magical about the improvisation in the in-person game. The fear of talking over people online (a real danger, in particular, because one can not read nonverbal queues about people starting to talk). I need to think more about how to compensate for online games.