User blog comment:Simplicityaboveall/Extremely Large Numbers 3/@comment-24920136-20160801224555/@comment-5529393-20160803050859

This seems to be a common mistake that people make, they contemplate a very large number or fast growing function, and are so astounded by it that they think it "must be much greater" than some famous number or fast growing function, even if they don't understand the latter very well. Many times I have seen people talk about some numerical construction that must exceed Graham's number or TREE(3), and the number turns out to be nowhere close. So one needs to be careful about that kind of thinking.

$$\Psi(\Omega_\omega)$$ is certainly a large ordinal, but on the scale of computable ordinals it is basically zero, just as any finite number is infinitesimal compared to infinity. So no matter how far up the computable hierarchy you go, you have no hope of beating the Busy Beaver function, provided you use any sort of algorithm to compute your function. And you have no hope of beating Rayo's function, provided that your construction is definable in set theory (a very loose restriction). That's just the way it is.

Also, I'm skeptical of the way you keep changing 10 with omega no matter what the expression involving 10 or omega is. Your "complete hereditary form" must be a single notation that expresses any natural number in terms of 10 in a unified fashion, and then gets you a transfinite ordinal when 10 is replaced by omega. It can't be "I can express an ordinal using any notation I want and then replace 10 with omega." Also, note that $$\Psi(\Omega_10)$$ is not a natural number so this can't be an instance of "complete hereditary form".