User blog comment:QuasarBooster/Laver tables in Python/@comment-34422464-20181211142242/@comment-35470197-20181211143653

No upper bound is known. Lower bounds are given by Dwerty, and the growth rate is greater than \(\omega+1\) with respect to a certain modified FGH.

On the other hand, the well-definedness of q(n) has not verified under ZFC yet. At least, it is known to be well-defined under ZFC plus the large cardinal axiom I0.