Yah, sounds like a custom game may be best for this. I could theoretically add code to automatically accept any players who join this game, and any characters they submit.
And you know there was a book I heard about that had a whole bunch of short stories, but they were all related because of the tavern. It was like a multidimensional tavern where these different fantasy worlds could walk through this tavern door and they were all in the same building. When they left, they all went to their respective worlds. Think of a bag of holding but the pocket dimension its connected to is connected to other openings. So this could be a fun idea for everyone to come in chat and go back to their respective worlds.
I wrote a blog post once about mashing systems together with no regard for stuff. Essentially, it could be up to the GM to adjudicate based upon respective probabilities of the systems in opposition. You would need a GM with a head for probability distributions and an understanding of the systems.
Multiple systems may be overly ambitious. What if we started off with one thread for D&D 5e since that seems to be the predominant system at the moment? I understand it is RP only, but done rolling may be necessary in order to add a random element to encounters/interactions.
Honestly it doesn't sound like we'd actually need a system for this, just the ability to have a forum in which any character sheet was allowed. Then you just post silliness from any of your characters. I can't imagine trying to workout "GAMEPLAY" would be much fun (but I could always be wrong).
7th Sea: Cannonball, Parchment, Rapier.
Numenera: Ancient super-powered technology that's shaped like a rock, Ancient super-powered technology that kind of functions like paper, Ancient super-powered technology that cuts things as an afterthought.