Here's a long list of open-source video games.
Things that I have enjoyed and played a lot (which tends to weight towards heavy replayability, and place little emphasis on graphical appearance, story, or even on approachability to new players):
- Battle for Wesnoth. A hex wargame with nice sprite art. Think Battle Isle, Advance Wars, Panzer General...but with characters that gain substantial stats and abilities and a fantasy story. Lots of scenarios made for it.
- Dungeon Crawl Stone Soup. This depends greatly on whether you like roguelikes or not (see /r/roguelikes for people who do). These are games that are typically turn-based, often played on a text terminal, played on a square grid, with some character cruising through a dungeon, finding items and powering up and learning that a "Red Potion" in this life is a potion of healing. DCSS is probably the best-balanced of these, and has some helpful automation to eliminate grinding.
- Lincity-NG. This is a Sim-City like game, though the mechanics are very different. Graphics roughly Sim City 3000ish.
- Urban Terror. Not open-source, but free (and runs on an open-source ioquake engine). There are far-newer FPSes out there; this is from the Quake 3 Arena timeframe, but I still haven't found something that replaces it; it's much-slower-moving than a lot of the newer FPSes. Open-source FPSes that are much newer than this that might have more appeal include Xonotic (aka Nexuiz) and Warsow. Neither is comparable in terms of bling to current-generation commercial FPSes, though.
- Aisleriot, the GNOME solitaire game. Never thought I'd play solitaire much, but I've enjoyed it; it's an easy drop-in, drop-out game. Eight-Off is a great solitaire game.
- ET: Legacy, a community-maintained version of Wolfenstein: Enemy Territory.