Lichess

Xonotic and Veloren are great.

Not a personal favorite, but many people seem to enjoy Mindustry.

0 A.D. - an Age of Empires style RTS that's actually really good, especially the background music for all the civilisations, the minor details on the structures, and the customisable AI behavior.

OpenTTD

In no particular order:

Online/LAN:

Xonotic: Good for online/LAN-play. UT-Style FPS.

OpenRA: Damn well good. RTS.

Warsow: Similar to Xonotic, but much faster. Damn good game. Sadly, defunct.

Sonic Robo Blast 2 Kart: If you like kart games and think they are all to easy, this is your choice.

Online/Split Screen/Couch coop:

SuperTuxKart: Damn fun, especially with a few add on tracks and good company at your place.

Hedgewars: Similar to Worms: Armageddon.

Battle for Wesnoth: Really fun once in a while. Neither the online nor the local experience is really "better".

Offline/Split Screen/Couch Coop:

Atomic Tanks: Worms on steroids.

Barbarian: Rocks. The OSS-Version is a tad bit obscure.

I didn't do VCMI, but Homm3 is one of my big local multiplayer favorites. I wait for the full inclusion of WoG before shooting it up. Also, as a young boi I really loved C-Dogs. The thing is now open source, check it out.

Red Eclipse or Xonitic for me

Linley's Dungeon Crawl aka Stone Soup. The most worked and polished old school roguelike there has ever been. Just in terms of game design, it is fascinating watching this project evolve. I have never seen a project make more fearless choices, and they almost always turn out to be winners.

http://crawl.akrasiac.org:8080/

Edit: OOPS! Not really multiplayer. There are some social features on the site that hosts it allowing you to watch other people's games and heckle them and stuff.

OpenArena

https://www.urbanterror.info/home/

https://www.openttd.org/

Warfork is my favorite by far. Also, their website lists a selection of other open source games at the bottom of the page.