Technically Exists

Googology proofs

Mu factorial proofs

The mu factorial has the same growth rate as the mixed factorial.

The mu factorial has the same growth rate as the Ackermann function.

The mu factorial has the same growth rate as  f_𝜔 in the Wainer hierarchy.

Fast-growing hierarchy proofs

Lower bound on the Wainer hierarchy up to 𝜔

Upper bound on the Wainer hierarchy up to 𝜔