My Top 5 Influential Games

Metal Gear (entire series, mostly Solid) -- that games can be as cinematic as movies
Lost Odyssey -- that stories can be very human, even in video games
Psi-Ops -- seriously! that gameplay can be so fun!
Shadow of the Colossus -- that gameplay can also be epic solitary
Mario Bros -- because that's where all games started after all...

if I may, I'll add a few other important games I've played:
Prince of Persia -- where you realize that yes, "game over" is pointless and it is best to let the player proceed through the story
Grand Theft Auto -- sandboxes rock
DDR -- yes, Dance Dance Revolution! way before the rock games today!
Flight Simulator -- computers and video games are great for real things too
Phoenix Wright -- great writing makes great games, even if gameplay is almost nil
and certainly many, many others...