Only the part for which we have well-defined fundamental sequences is constructive. As far as I know, there is no such system of FS defined up to PTO(Z_2), the Proof Theoretic Ordinal of second order arithmetic, while growth rate at that ordinal can be programmed in under 42 bytes.
> waaaaay beyond current known BB bounds
I have to disagree here. The Proof Theoretic Ordinal of ZFC + infinitely many inaccessibles can be reached with a program under one kilobyte in size, and that is already extremely high up into the FGH.