Gaming

OpenRA has open source slightly modernized versions of Red Alert, Tiberian Dawn and Dune 2000. There's also Tiberian Sun-like and Red Alert 2 mods for it :)

Endgame: Singularity is a fun (and sometimes very frustrating) small strategy game worth checking out.

The Dark Mod is basically Thief but really well done with a solid amount of (fan) missions to choose from. Speaking of I should play some of it again.

Puzzle

• Rocks'n'Diamonds - Digging for diamonds while avoiding boulders and enemies
• Enigma | github - Marple puzzles (and much more) inspired by the classic game Oxyd
• GNU Robbo - Solve rooms (puzzles) with your robot

Space

• Oolite | github - an open-world space opera (inspired by classic game Elite)
• Kobo Deluxe - 2D scrolling space shooter, destroy star bases with your ship
• Open Tyrian - Arcade-style vertical scrolling shooter

I still jump into Quake 3 DeFRaG sometimes. It's a movement game, probably one of the earliest examples of a game / mod where the primary focus is on movement as opposed to shooting or some other mechanic. Its mere existence heavily inspired a lot of source engine movement communities like surf, bhop, kz, etc.

Here's a good video showcasing what DeFRaG movement looks like. Each map is ran on two variations of the physics. One is VQ3 (vanilla quake 3) and CPM (Challenge ProMode Arena, which was originally a Quake 3 mod that heavily altered the physics to make it more fun for professional play). While there's no one DeFRaG game, there's a shitload of open source mods to the original game that modify the physics and allow you to join servers running DeFRaG maps. Some of them even hook up to the online database allowing you to register times without recording your runs and submitting them the traditional way. There's usually a new open source defrag mod rolling out every 3 years or so, and each one builds upon the successes of the previous one. The one I currently use is iDFe which I think came out around 2018 or so? I'm pretty sure there's a new one, and there are other games in the works in both modified Source Engine and Unreal Engine 4 that are looking to emulate the VQ3 and CPM physics for modern engines.

The community is pretty dead, but it rises from the ashes every few years for DFWC (the video I linked above was DFWC 2017). You can always play any one of the thousands of maps in singleplayer too.

edit: I should add, it's not totally open source. You still need to own a copy of Quake 3 (or visit the high seas, at least) because each DF mod requires you to verify your ownership of the game by using a valid CD key. There is a "universal" developer CD key, but I don't remember what it was. When you download defrag you're effectively downloading quake 3, but it's bricked if you don't have the CD key.

Xonotic. I even got a server in my region!

If reverse-engineered Nintendo games count, then Super Mario 64 and Pokemon Emerald whose engine is technically open source.

EDIT: Minecraft Beta 1.7.3 has been reverse engineered multiple times too.

OpenTTD is a great open source clone of the transport tycoon game. And boohu is an awesome ASCII coffeebreak roguelike.

Simutrans. It's old but really good. It's really easy to lose 10h or so in it before you notice the time.

OpenTTD is my drug. I'm addicted

There are a few open source game projects I follow. I suppose the most famous one is the Freespace 2 source code project. Although it didn't start open source, the original devs open-sourced it later. It has great support, and a great modding launcher called Knossos. To play the game, even with the source code, you either need the original disks or a copy of the installer from GoG, but it's really cheap. Getting it working on Windows is pretty easy, but Linux is only slightly more complicated. (Fortunately, there's a new launcher that makes it way easier).

If you're interested, let me know.

If you didn't already suggest it, I would have suggested Veloren!

Pokemon Emerald 😉

Battle for Wesnoth is a personal favorite of mine that I haven't seen mentioned elsewhere in the thread. It's a pixel art, hex-grid turn based strategy game with fantasy armies. It's mixed with mild RPG elements, and there's tons and tons of campaigns and fanmade content accessible directly through a mod browser in game.
The other games I enjoy that I see being mentioned here are in the same vein. BAR, Warzone 2100, OpenTTD. And the open mods based on the STALKER engine, like Anomaly. I love those.

Dungeon Crawl Stone Soup for me. A classic roguelike with active community and regular updates as well as several forks. It also has a good amount of playable races. From classics like elfs or orcs ~~and dwarves~~ (RIP Mountain Dwarves you were too good for this game) to funky races like Vine Stalkers (sentient parasitic plants) and Armataurs (centaurs but with armadillo parts instead the horse ones) to cats and octopuses.

And the gods you can worship in the game are also very diverse. How about a slime god? Or someone who wants you to wear as much cursed equipment as possible in exchange for knowledge? Or maybe the one who wants everything to move veeery slooow and rewards you for killing mobs that are faster than you?

OpenRCT2, eDuke32, the various Doom and Quake games

Also ScummVM is open source and supports many games

I like supertuxkart and minetest

InfraArcana was a little bit tough but is a very good roguelike

It's old but I always really liked WARSAW it was/is a quake3 port but with more cartoony outline graphics similar to Borderlands style.

Veloren

For me it's CDDA, a quite realistic and deep apocalyptic survival game with very frequent updates(on the experimental branch)

0.A.D., Veloren and Xonotic are my favorites. Also Sonic Robo Blast 2 is freaking awesome

I'm surprised to not see Flare RPG mentioned. It's nothing groundbreaking on its own, but it's actually got two nicely fleshed out campaigns, and tooling for people to make new ones. It's a nice, fun, FOSS single-player RPG, and it's great if you want that old fashioned Diablo feel.

Another vote for Endless Sky here as well, it's just excellent, and surprisingly expansive.

osu!lazer

Surprised it's not mentioned here, but Bzflag.

Super fun tank shooter game that doesn't take much to run, and reminds me of a cross between the very old bolo game and Mario kart's battle mode.

Doom :)