Cave Story is the single finest free game i have ever played

Path of Exile is a good Action RPG if you're into that. Bit of a learning curve and you'll probably want to follow a guide if you want to do endgame stuff.

Trackmania Nations Forever is free on steam if you like to do loops and go fast

Recently tf2 went f2p I believe

cpma :D

http://nquake.com

Aside from cave story this is a pretty good CLASSIC

Paladins is pretty fun

Killing floor isn't free but it's dirt cheap during sales (since there was a sale recently, you can probably get a copy on tf2outpost for 1 key). It's an amazing game that you can play alone, with randoms or with friends and you can easily get dozens if not hundreds of hours from it

Paladins is a great F2P game

But I literally came to this thread just to shitpost about TF2

WARMODE

fistful of frags

I actually played in its small competitive league for one season, it's surprisingly fun as an arena shooter.

Warframe, gameplay is really fucking fun but there's micro-transactions and that sort of stuff but you can go through the entire game without paying anything.

Dungeon crawl Stone-Soup is a great rogue variant.

minesweeper, solitaire, portal story mel is free if you have portal 2.

Since you enjoy TF2, I'm going to assume you get off on bad graphics. So I'd recommend you check out the following:

-ADOM (roguelike)
-Battle for Wesnoth (TBS)
-Dwarf Fortress (Warning: this game causes autism, do not play unless you don't mind having dreams about the best ways to deliver minecarts full of weapons unto your enemies. It's basically Elder God Minecraft)

chess.com

Warsow
CPMA (Join the official CPMA Discord and you will find a link to download it)
Super Crate Box
nQuake as #7 suggested

If you like 2D action/adventure games, Daniel Remar's games are very good, particularly Iji and Hero Core.

DotA2

Second Path of Exile, it rules. Only keeps getting better every major update, IMO. Duelyst is also a pretty nice F2P card game that seems quite complex and P2W-free, though I didn't play it too much.

As for single player games, I second Cave Story and Iji, also:

• Nitronic Rush, a pretty neat futuristic arcade racer (being commercially remade as Distance)
  
• Perspective, a surreal puzzle platformer where the 3D camera's perspective changes the geometry in a 2D platformer (just play it, you'll get it)
  
• Space Funeral, a pretty short and easy RPG Maker game that's really all about the surreal atmosphere and sweet mixtape of a soundtrack (and metafictional elements). Looks like an EarthBound tribute but isn't really. The fan sequels are mostly mediocre, but they're there if you want more. Stay away from SF4 though, it's disgusting. Might want to check out other games by the same guy as well, like Murder Dog IV (it's hilarious).
  
• Gun Godz, an intense hour long 2.5D FPS that feels like Doom on amphetamines. Some replay value in mastering the levels until you get perfect grades. From the makers of Nuclear Throne (and Super Crate Box, which was mentioned above but I'm not really a fan)
  
• Sluggish Morss, a surrealistic... something. I don't know. It's cool though. Sequel is very different but is as interesting. Second sequel is commercial.
  
• Tag: The Power of Paint - you've played Portal 2, but have you played it without portals and with a mind bending sticky gel instead? Probably not. Takes an hour to beat. Warning though, very low nonadjustable FOV.
  
• Soldat - not sure how the community is nowadays, but it's still modestly fun with bots.

Also emulators.