Pfft, it's much smaller than the number of atoms in the universe. Then again, after I read how to calculate Graham's number, I'm not impressed by much.
The fun bit about TREE(3) is that the sequence TREE(1), TREE(2), TREE(3) goes :
TREE(1): 1
TREE(2): 3
TREE(3): explosion
n(4) is another fun one. n(3) is less than Graham's number, which itself is roughly A^64(4). n(4) is about A^A^(187196)(1)
E.g. http://mathoverflow.net/questions/93828/how-large-is-tree3