Free Code Game Engines
Crystal Space - LGPL license, programmed in C++
3D Open World
WorldForge - a 3D engine developed by the WorldForge project, under a combination of GNU GPL, LGPL, and other GPL-compatible licenses
Ryzom - a 3D engine (client and server) released to the FSF by Winch Gate Ltd in 2010, under GPL/AGPLv3.
Meridian 59 - a 3D MMORPG engine developed as proprietary software by a commercial operator, and released into community development under the GPL in 2012. Only runs on Windows as of March, 2016 (might work in WINE?).
3D First Person Shooter
Cube - developed under the zLib license, a non-copyleft free software license
DarkPlaces - forked from the Quake engine, both of which are licensed under the GNU GPL
Doomsday and Gloome - rewrites of Doom (Doom engine released under GNU GPL by ID Games)
Torque 3D - "MIT'" licence, oriented towards Windows and browser games, although it also supports GNU/Linux and MacOSX.
Lugaru - developed in the C language by Wolfire games and released in August 2009, details about this game are sketchy but the screenshots look epic! The codebase is orphaned, but some amazing games could be build if new developers were found. Wikipedia states that the code, now hosted at BitBucket, is under GPL, and the game data is proprietary. There are Debian packages but not sure which repo, so that doesn't help.
Irrlicht - a C++ 3D Engine under the zLib license, and used a wide range of games projects including..
Minetest, a MineCraft-like engine forked from Irrlicht, and released under LGPL 2.1
2D and 3D
Godot - a full game IDE developed in C and C++ by an Argentine game studio and released under the Expat ("MIT") license. Used in a number of libre games including DynaDungeons (GNU GPLv3), and Tanks of Freedom (Expat)
Top Down Realtime Strategy
Pyrogenesis - Developed by Wildfire Games for 0AD, and released under the GNU GPL
Stratagus - Used in BOS Wars, Doom Wars, Wargus, and Wyrmsun, developed under the GNU GPL
Spring - Developed under GPLv2, and used in a wide range of games, some with libre art from the historical (Spring: 1944), to the futuristic (Zero-K), to the abstract (Kernel Panic), some with proprietary art (Star Wars: Imperial Winter).
Top Down Turn-Based Strategy
Battle for Wesnoth - GPL
OpenBOR - intended for Street Fighter side-scrolling style games, this "BSD" licensed engine is written in C and uses OpenGL, Framebuffer, and SDL
OpenOMF - released under an "MIT" license, this is a remake of a DOS game, One Must Fall 20
2D Side-scrolling Platformer
PGE (Platform Game Engine) - a project to create a GNU GPLv3 engine to support Super Mario mods like SMBX
SGE (pronounced "sage") - a 2D engine originally developed for a game called Stellar, currently used by PaceWar
Phaser - This Expat ("MIT") licensed engine has been used for a wide variety of games, including Angry Birds Space
Turbulenz - Another Expat ("MIT) licensed engine dependent on WebGL, used for a range of 2D and 3D animated games on the Turbulenz site
Music and Dance
Stepmania - a DanceDanceRevolution style game, whose engine is developed under a non-copyleft free license
Frets on Fire - A live guitar playing game, whose engine, based on PyGame, is developed under the GNU GPL
UltraStar - A SingStar-style game developed by SterGames in Kylix/ Delphi, under GNU GPL 2, with ports for GNU/Linux and OSX in C++. Unfortunately, the original developers moved to proprietary development (and appear to have gone bust around 2012), but source code is still available on SourceForge, and a number of forks and rewrites exist such as UltraStarDeluxe.
Performous - An engine for games based on singing, instrument simulation, and dance, licensed under GNU GPL 2. Began as a C++ rewrite of UltraStar, targeted at GNU/Linux.
Repositories of games which have either released their source or been developed as free code
Internet Archive Game Source Code Collection
Wikipedia List of Open Source Video Games
Free Software Directory list of Free Code Games
Lee Reilly's list of game-related projects on GITHub