Indie game developer Brendan Chung (http://www.blendogames.com) used a very lightly modified Quake Engine for his two short-form experimental narrative-based games, Gravity Bone and Thirty Flights of Loving. (The first is freely downloadable, the second you have to pay for). He's also working on a new game, Quadrilateral Cowboy, using the Doom 3 engine - apparently the gameplay might be more conventional for this one, though hopefully not too conventional...

The Quake 3 engine is used for fairly well-received open source arena shooters like Tremulous, Xonotic, and Nexuiz.

The Dark Mod is a Doom 3-based Engine for fan-made games in the style of the Thief Series

You also have a few ports of engines to cope with modern windows versions or non-Windows operating systems or just updated gameplay (Jagged Alliance 2 straciatella or 1.13 springs to mind).

The GPL version of the Quake II engine has been used to make a few games:

https://en.wikipedia.org/wiki/Quake_II_engine#Games_based_on...

People port games like Doom and Quake to every new device and architecture. They fix bugs and integrate modern technology.

There are a few out there. Xonotic is one.