I like the idea of doing it in order of exposure, and adding a bit of notation to explain why it's influential to me, so I'm going to write my list that way. There are probably other games that have had a significant impact on me, and there will be a list of honourable mentions at the end.
1. Dragon Warriors
2. Middle Earth Role Playing
3. Cyberpunk 2020
4. The World of Darkness (notably Werewolf the Apocalypse and Mage the Ascension)
5. Warhammer Fantasy Roleplaying
7. Ghost Dog: Way of the Samurai
9. A Penny for My Thoughts
This is one of those games that I didn't think much of until I played it. It is written in the form of an in-world artefact, and the game play is very ritualised, but like 3:16 it does one thing and it does that one thing really well. I've adapted this game so many times over the last few years, using it as a character generation system to develop backstories, and using it to weave communal narratives that help to establish a world.
Teenage Mutant Ninja Turtles and Other Strangeness.
Chill 2nd Ed
Legend of the Five Rings