Can there be/ has there been created a meta-theory that recognizes the existence of all objects whose existence is derivable from (or just "true in") each axiomatic system analyzable by the metatheory? From my (naïve) position, it seems like you could have ordered pairs in the metatheory, where the first place is an axiomatic system and the second place is an object of the system.
My interest in this is partly because it would make the metatheory "huge" informally, and if for instance we did this meta to 1st order theories, our metatheory would already contain (ZFC+L, l) where L is an arbitrary large cardinal axiom, and l a large cardinal implied by the axiom. In this way, we could in a sense beat the large cardinal naming game by not playing.