When does Busy Beaver surpass TREE(3)? I saw a question which asked, "When does Busy Beaver surpass TREE(n)?" I am asking a somewhat different question, about a specific value of TREE. I know that TREE(3) is an unimaginably vast number, far more than even Graham's number. I would be satisfied with even a good upper bound when Busy Beaver surpasses TREE(3).
 A: As far as I know, there is no known upper bound for $TREE(3)$. To have at least a slight idea how huge this gap is , you should study the "fast growing hierarchy".
Graham's number has level $\omega+1$.
Conway chains far surpass Graham's number, already $4\to4\to4\to4$ is vastly larger. The level of Conway chains of length $n$ is about $\omega^2$
Bowers arrays far surpass Conway chains. $[3,3,3,3,3]$ is already inexpressible with Conway chains.
Even Bowers planes only reach $\omega^{\omega^2}$-level and with $3$ dimensions $\omega^{\omega^3}$.
This is still far away from the $\epsilon_0$ level and this is FAR surpassed by the $\Gamma_0$ level.
And even this level is FAR too low for $TREE(3)$.
I think you begin to understand that $TREE(3)$ is not just another league , the difference in magnitude is barely comprehensible even in googology-standards.
The busy beaver function however should surpass the TREE - function relatively early, but I have no idea where. I think, it does it below $n=100$.
