explicit upper bound of TREE(3) TREE(3) is the famously absurdly large number that is the length of a longest list of rooted, 3-colored trees whose $i$th element has at most $i$ vertices, and for which no tree's vertices can be mapped to the vertices of a subsequent tree preserving color and inf relationships.

Some lower bounds of TREE(3) have been proven. Among them are $A^{A(187196)}(1)$, where the superscript denotes function iteration and $A(x) = 2\uparrow^{x-1}x$ (using Knuth up-arrows) is a version of the Ackermann function. More bounds appear in this Wikia article and here. These bounds are rather unsatisfying because there is no indication of how tight they are.

The only upper bound of TREE(3) that I have seen (other than, trivially, TREE($n$) for $n &gt; 3$) has a similar derivation as a longest sequence of a more general type of graph; I can't find a reference at the moment. This upper bound is also unsatisfying because it is non-constructive.

Does anyone know of an explicit expression that is an upper bound of TREE(3)? Or is it so large that there is no hope to construct one?
$\ 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.
TREE(3) is known to be finite, but few finite upper bounds have been proven that do not rely on constructions such as TREE(n) and n>3. 

As for lower bounds, saying $g_{64}$ (Graham's number) is a lower bound for TREE(3) is like saying 1 is a lower bound for Graham's number. Even $g_{187,196}$ has proven to be an EXTREMELY WEAK lower bound for TREE(3). 

See 
https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem 
and 
https://www.youtube.com/watch?v=3P6DWAwwViU.