My list will be a little different -- because the most influential ones for me are the games that made me want to become a game developer...so some of this stuff is OLD.
An early arcade videogame that gave rise to similar side-scroller shoot-em-ups like Super Cobra. For whatever reason, this is the game that really got me thinking about making games. I remember drawing the maps for levels, thinking up new weapons and enemies... I was probably 7 years old.
Adventure (Colossal Cave)
Played this on a DEC-20 mainframe. The first game that showed me that computer games could let you explore (very much in your mind) a completely new space. I remember the huge plotter-printer map of interconnecting rooms, along with places for loot and obstacles. Another game that inspired me to make games.
This was a simple text game on the C-64 which let you allocate limited resources (food and money) to peasants and army in a feudal kingdom. Misspending resources could be apocalyptic. The game was simplistic, but it inspired me to make the first commercial game I ever made: Space Empire, a multiplayer "door" game for the Atari ST, which dealt with a similar resource management game mechanic but took place in a science-fiction world--and against other players. I released it for shareware at $10 a pop, and at age 13 I think I raked in about $4000 (and believe me, at that age it was a life-changing amount of money!) In 1987 I ported it to the VAX and the Amiga, and then released the source code. Numerous ports followed (including the PC) and tons of derivatives were created. A lot of that multiplayer door goodness could be traced back to my own Space Empire--which I in turn created based on the simplest of C64 games.
The multiplayer MUD on the GENie network circa 1991. This is the game where I met my future wife, and inspired me to create Legends of Future Past, a commercial fantasy roleplaying game I launched in 1992 and ran until 2000.
Knights of the Old Republic
The first game that made me feel sad for taking certain actions as an evil character. Beyond that, it was just a damn fine experience in every way. Star Wars that was even better than Star Wars. The sequel didn't quite measure-up.
Am I done five already? Wish I had made this a list of 10!