These are quick games that a ton of people can play at once, and there's plenty of opporitunity to join mid-game session.

http://store.steampowered.com/app/212480/Sonic__AllStars_Racing_Transformed_Collection/ (Sonic All Stars can have up to 10 players)
http://store.steampowered.com/app/252950/Rocket_League/ (Rocket league can be up to 4v4)
http://store.steampowered.com/app/236390/War_Thunder/ (War Thunder can have a lot of players in a private match. Harder to organize. It's also free-to-play.)
http://store.steampowered.com/app/4000/Garrys_Mod/
http://store.steampowered.com/app/286160/Tabletop_Simulator/ (Tabletop games may or may not be quick. It depends on the game.)
Any Jackbox game - Only the host needs to own the game.