User blog comment:Edwin Shade/There is No Limit To Googology/@comment-31663462-20171120001610

Your idea reminds me of something similar. Imagine using ZFC as our initial system. It has an unknown but very large proof-theoretic ordinal, which is analogous to your $$n$$. But we can go farther. Consider including an axiom that states at least one inaccessible cardinal exists. Woohoo, now we've got something akin to your $$n'$$. And then we can add in an axiom that states at least two inaccessible cardinals exist, etc.

But little do we realize, the calculus of inductive constructs has a proof-theoretic ordinal equivalent to ZFC + countably many inaccessible cardinals. Wham! There's a limit you can't surpass quite easily, using the above schematics. I believe Saibian means something similar. There will be a system even further beyond your current system + all the axioms you'll ever think of.