How many Clubs you can have in a game?

I don't think there is a limit to the number of clubs if there is it would be a pretty high number.

A single Sim can be involved in up to three clubs. The total amount overall for clubs in a single save; I don’t think there is a limit for that.

I don't think there's a limit, just empty clubs are culled and replaced with new ones that scan for sims with the relevant skill/traits to replace them

For some reason I think it's fifty. I feel like I read that somewhere way back when Get Together first came out, but I'm not certain.