Some games I've enjoyed 10-20 years ago. Most of this list is ancient and
I don't necessarily recommend all games any longer. Some random new games have
been randomly added.

Newer-ish games

Katamari Damacy series                  Extremely fun, strange and stylish, irresistible music
Mercenaries: Playground of destruction  A fun war action game with lots of freedom
Rock Band 3                             Pro keys is extremely fun
Burnout series                          Great "racing" game, the crash mode is great fun
Dead Rising                             Rather silly but great zombie action game
Grand Theft Auto series (III-SA)        Free-roaming action, great fun, both for playing and råning
Saints Row series                       GTA-alike series that went crazy while still being extremely fun
Mafia                                   Similar to GTA, but less open and more focused on story
Railroad Tycoon II                      It's even better than the Sid Meier original
Hitman series                           High quality game stealthy action
The typing of the dead                  Hilarious zombie horror typing game
Dance Dance Revolution (Stepmania)      A brilliant form of exercise, step on arrows in time with the music
Alpha Centauri + Alien Crossfire        A great follow-up to Civilization
Castlevania (SotN, DoS, PoR, OoE, ...)  Great exploration-platform games
Super Panda Adventures                  Gameplay similar to several Castlevania games, silly and fun
Dragon's Dogma                          RPG with fantastic combat
Fallout series                          Great post-apocalyptic RPG with lots of content
Risen series                            Great classical RPG
Exit series                             Puzzle-heavy platform games, although with a rough interface
SpaceChem                               Pretty much the best puzzle game ever
Chip's Challenge 2                      The legendary puzzle-action game which finally got released
Deus Ex series                          Great "stealth-RPG"
Spelunky                                Great platform game with procedurally generated levels, has a lot of depth
Crypt of the Necrodancer                Extremely challenging rhythm roguelike with even more depth
Long Live the Queen                     Visual novel/skill-building game with excellent writing and immersion
Lost Marbles                            Great Marble Madness-ish game with racing and puzzles
Euro Truck Simulator 2                  Very relaxing and visually pleasing game where you drive around and transport stuff
Cookie Clicker, Clicker Heroes          Watch numbers increase. Warning, not for everyone

Small and fun games

Bricks                                  Classic block sliding game, freeware
BD4                                     Fantastic remake of Boulderdash and Emerald mine with online
                                        high scores, freeware
Tile world                              A remake of Chip's Challenge, freeware

Classics

Civilization 2: Multiplayer Gold ed.    The best version of the best strategy game ever made
Colonization                            One of the best strategy games ever made
Maniac Mansion, ...,  Grim Fandango     The best adventure games ever made, period.

There are really too many classics that I bother to mention all of them.