openTTD ^[1] is one of the best, and most stable open source games out there. Since version 1.0, it is completely independent from the files for Transport Tycoon Deluxe, the game it was cloned from.

Roguelikes

• Nethack ^[1] - an oldie but goodie. Not updated much anymore; the newest version was released in 2003 and the last ports were updated in 2009.
• SLASH'EM ^[2] - Nethack with everything and the kitchen sink. Also, lightsabers.
• Dungeon Crawl: Stone Soup ^[3] - One of the most fair games on this list, in terms of the likelihood of dieing instantly. Nice graphical interface, too. Start here.
• Angband ^[4] - Very popular, lots of derivative projects. Emphasis on slowly venturing into the unknown.
• Ivan ^[5] - Odd banana theme. Unfairly difficult. You can rip your opponents limbs off and beat them to death with them.
• Spelunky ^[6] - A melding of genres; a sidescrolling platformer with roguelike theme and randomly generated levels.

The Humble Indie Bundle ^[1] have recently open-sourced 4 of their games.

I believe you can still buy them in binary form, but the source is there if you want to play around with it.

4X (Civilization-like)

• Freeciv ^[1] - A clone of the Civilization games which has been actively developed for many years and, IMHO, is still just as addictive as ever.
• C-evo ^[2] - A clone of Civilization II, with an eye towards fixing imbalances and making the AI more interesting.
• FreeOrion ^[3] - Inspired by the "Master of Orion" games.
• Thousand Parsec ^[4] - A framework for making 4X games set in space. There are several playable rulesets linked on the site.
• Star Trek: Supremacy ^[5] - Appears to be a reimagining of the old Star Trek game. The source code is located here ^[6].

If you're interested in turn-based strategy, Battle for Wesnoth ^[1] is quite a lot of fun, and the open-source community is constantly adding new campaigns, or if you prefer, there is a multiplayer option available. It is under active development and every year has a Google Summer of Code ^[2] participant involved.

It's available free on PC, *nix, and Mac, and for a nominal fee on iOS (though it is something of a battery hog).

Realtime Strategy

• Warzone 2100 ^[1] - Originally released in 1999 as a commercial title, the source code has been opened and all assets are included.
• Spring ^[2] - While the project started as a remake of Total Annihilation, Spring is now a general RTS engine used in a variety of games.
• BOS Wars ^[3] - Somewhat reminescent of Command and Conquer. The successor of the Stratagus ^[4] engine.
• Glest ^[5] - Two warring factions of magic and technology.
• Globulation 2 ^[6] - Has an emphasis on macro play.
• Dark Oberon ^[7] - Heavily inspired by WarCraft II, DO has the distinction of being the only RTS I know of to use claymation graphics.

Since i've not seen any FPS, I would like to remind you a few of them

• War§ow ^[1] a fast-paced FPS with a nice cell-shaded look
• Cube ^[2]/Sauerbraten on which is based AssaultCube ^[3].
• OpenArena ^[4] pure deathmatch, Quake-engine based
• Nexuiz ^[5]
• UrbanTerror ^[6], a so-called Hollywood tactical FPS, which adds to Soldier of Fortune realism a bunch of spectacular moves
• Tremulous ^[7], which opposes humans to aliens, and adds some RTS elements.

Pingus ^[1].

[From their website]

Pingus is a free Lemmings-like game covered under the GNU GPL. It features currently 22 playable levels and runs under a wide variety of operating systems (Linux, Windows, etc.)

It has, unfortunately, not been updated in a while, but it's certainly playable - SuperTux ^[1]. It's available for Windows, Linux, (I think) OS X, and it's even been ported to the Wii ^[2].

If you do play it, make sure to run the development version - it's much better and just as stable. (This might be available in your Linux distro as supertux2.)

SuperTuxKart ^[1].

[From the website]

SuperTuxKart is a Free 3d kart racing game; we want to make the game fun more than we want to make it realistic. You can play with up to 4 friends on one PC, racing against each other or just try to beat the computer.

FreeDink ^[1]: funny and plenty of mods. There's a great community out there with the mods (that you'll maybe want after you've finished the game): just search for DFArc ^[2].

If you like card/puzzle games, I recommend MahJong ^[1] and Shisen Sho ^[2]

Pioneers ^[1] is an open-source clone of the Settlers of Catan board game.

Open Red Alert ^[1] is an open clone of red alert 1. But it's not the most stable at the moment.

If you like roguelikes but can't stomach the unfathomable complexity of Nethack, try POWDER ^[1]. There's even a tutorial and ports to several consoles and phone platforms, including PSP, DS, GP2X, PS3, Wii and iPhone.