There doesn't seem to currently be a way to grab the game_id other than copypasting it from the console, or brute-forcing our way through the matchmaking api's server list. It'd be neat if we could access it via the Modding API, so we can:

a) request it from within the game via a /c command
and
b ...

An event that fires whenever a player says something in chat (or uses commands, although a separate on_player_command might be good for that)

...as far as I can see, nothing like that exists so far. I'm trying to do something related to logging and this would help a lot!