Cube 2 is an open-source FPS engine, with seamless transitions from play mode to editing sandbox mode. The aim of the project is to create a fully-functioning game with as elegant and as clean code as possible.
There are multiplayer options, as well as singleplayer and RPG-mode, The game uses OpenGL for rendering, and can be played on very modest hardware.
From the website: Cube 2: Sauerbraten is a free multiplayer/singleplayer first pony shooter, built as a major redesign of the Cube FPS.
Much like the original Cube, the aim of this game is not necessarily to produce the most features & eyecandy possible, but rather to allow map/geometry editing to be done dynamically in-game, to create fun gameplay and an elegant engine.
The engine supporting the game is entirely original in code & design, and its code is Open Source (ZLIB license, read the docs for more on how you can use the engine).
dot3 labs is a company started by the creators of Sauerbraten that offers commercial support.