explicit upper bound of TREE(3)

$\ TREE(3)\ $ is far above the $\Gamma_0$-level of the fast growing hierarchy.

See here how insane large it is : https://sites.google.com/site/largenumbers/home/appendix/a/numbers3

I am not sure whether an upper bound is known. Nested Ackermann-functions do not come near to $\ TREE(3)\ $ at all. Conway chains and even Bowers's $3D$-arrays are still MUCH MUCH MUCH too small.

After discussing the problem with Andreas Weiermann, it appears there are no good upper bounds to TREE(3). One can derive a good asymptotic bound to TREE(n) on the order of $F_{\theta(\Omega^\omega \omega,0)}(n)$, however this does not in and of itself prove any bound to TREE(3).

I suppose it seems unlikely, given the asymptotic upper bound for TREE(n), that TREE(3) would exceed say $F_\alpha(3)$ for $\alpha$ much larger than $\theta(\Omega^\omega \omega,0)$, but this is speculative.

The underlying rationale will be as follows. The Tree(n) question can be translated into a problem of the length of effectively given bad sequences. I think this goes directly back to Friedman. Such effectively given bad sequences can be mapped into effectively descending sequences of ordinals using the reification technique. Effectively descending sequences can be bounded by the Hardy hierarchy (by say an old MLQ article by Buchholz, Cichon, Weiermann). This will also give a reasonable concrete upper bound for n=3. The problem in writing this up is that until now according to my judgement very few people were interested in it. Moreover the material would be considered as "folklore" and so a publication might be turned down.