I should perhaps have asked this question to Harvey Friedman himself... Anyway, a reasonable answer is given on [the talk page for Wikipedia article on Kruskal tree theorem][1], where one can find this quotation (from H. Friedman himself)

>Also, numbers derived from Goodstein sequences or Paris/Harrington Ramsey theory, although bigger than n(4), are also completely UNNOTICEABLE in comparison to TREE[3].

To clarify my remark "any number expressible by iterations of n" in the question (and answer Dylan Thurston's comment), I meant that TREE(3) is bigger than expressions like $n^{n^{n(100)}(100)}(100)$, say, where again exponentiation means iteration, and the whole expression has no more than $n(4)$ symbols; this would indeed be true if, for instance, as suggested, we had TREE(3)$>f_{\Gamma_0}(n(4))$.  For reference, I should have added that TREE(3) is the incredibly (at first, or even second look) large answer to the question:

> which is the length of the  longest sequence $(T_2,T_3,T_4,\dots,T_n)$ of labeled trees such that $T_k$ has at most $k$ nodes labeled $a$ or $b$,   and $T_i$ is not a subtree of $T_j$ for $i < j$?


  [1]: http://en.wikipedia.org/wiki/Talk:Kruskal%27s_tree_theorem#Correcting_TREE.282.29