What is the smallest number $n$ for which $bb(n)>f_{\epsilon_0}(5)$ is known?

Question

It is known that $bb(23)$>Graham's number (I do not remember exactly, but $bb(21)$ could already be larger).

But what is the smallest number $n$, such that $bb(n)>f_{\epsilon_0}(5)$ is known ?

Here :

Milton Green's lower bounds of the busy beaver function

it is mentioned that $\Sigma(41,3)$ is much greater than $f_{\epsilon_0}(5)$, but I am looking for a binary machine.

Answer 1

Wythagoras created an 85-state Turing machine that produced more than $f_{\varepsilon_0}(1907)$ ones, listed here. I believe it is based on an implementation of the Kirby-Paris hydra.