id Software open sourcing the Doom* and Quake 1..3 engines was very generous and all engines have since seen improvements through public contributions. Unfortunately game data is proprietary, however there are many open games using these engines.
FreeDoom, Nexiuz, Xonotic, Warsow, Urban Terror, World of Padman, Tremulous, OpenArena are popular examples of wholly free games based on id's engines, though some of them may be more actively maintained than others.
Quake3e is my favorite, despite requiring a copy of the game.
Xonotic is absolutely amazing multiplayer. Extremely competitive and fast paced. There isn't a large number of online players but I can always find a Deathmatch or Team Deathmatch to burn 15 minutes on. The weapons are excellent, like a mix of Quake and Unreal Tournament. I play it almost every day.
Edit: if you download it and install it, be aware that there's a script buried in 'misc/tools/rsync-updater' that updates to the latest autobuild. The last official release, 0.8.2, is really old.
In the same vein, there is the Freespace source code project (https://www.hard-light.net/about/freespace) which is the continuation of Descent: FreeSpace 2 after Volition opened their source code in 2002. The original game data is not open, but there's been a lot of community effort into creating new content and updating the original content, to the point where there's even been community translations of the original voice acting.
I thought about including it as a suggestion, but the modding community doesn't seem to care about licensing. The code is all properly licensed (Apache/MIT), but FAFAICT none of the content mods come with a license file, which means it's all free as in beer, not free as in speech.
Not all of these are wholly free games. For instance World of Padman's resources (models, textures, ...) are proprietary. I think at least some versions of Nexuiz and Warsow had same issue (not sure what is current status).
Freedoom is an IWAD that's meant to replace all of the assets in the original game files with free versions. You can then use it to play any addon level, like the massive total conversions or level packs that win awards every year. A lot of modern level packs use a free texture set called otex that's a massive/near complete replacement of doom textures too. Sure you won't be fighting exactly the same monsters but Freedoom is surprisingly high quality and worth checking out.
And ultimately the original games go on sale for digital purchase for a few dollars all the time so they're not a huge expense to eventually get and then play all the free levels people create as they intended.
Sorry, I'm not sure how this is related to my comment. All I wanted to say was that the examples the person I replied to listed (xonotic etc.) are good games, just not popular.
FreeDoom, Nexiuz, Xonotic, Warsow, Urban Terror, World of Padman, Tremulous, OpenArena are popular examples of wholly free games based on id's engines, though some of them may be more actively maintained than others.
Quake3e is my favorite, despite requiring a copy of the game.