How Big would "Graham's Tree" be?

Question

What if in Graham’s Number every “3” was replaced by “tree(3)” instead? How big is this number? Greater than Rayo’s number? Greater than every current named number?

Answer 1

No. If you replaced all the $3$'s in the construction of Graham's number with $\operatorname{TREE}(3)$, the resulting number would be smaller than $g_{\operatorname{TREE(3)}}$ where $g_n$ denotes the $n$th number in Graham's sequence with $g_{64}$ being Graham's number. This is much much smaller than $\operatorname{TREE}(4)$, for example.

Answer 2

No, Rayo's Number is just too big, imagine a Googol symbols in the first order set theory, you cannot express it, why? because even writing down a symbol per Planck time (5.39 x 10^-44 seconds) it would still take about 10^48 years, and another problem is the space, the number of particles in the observable is about 10^80, a Googol is 10^100,and bigger than any named number except infinity, infinity is NOT a number, this number would be bigger than TREE(3), but smaller than TREE(4), which is much bigger than TREE(3), and then if you pick a number like TREE(g(64)), or TREE^TREE(3)(3),(where TREE^2(3) = TREE(TREE(3))) or a number even MUCH bigger than Rayo's Number like FOOT^10(10^100) or Fish Number 7, this "Graham's TREE(3)" wouldn't even be 0 against them.