TITLE: Lower bounds for bb(7) and bb(8) wanted
QUESTION [6 upvotes]: The busy beaver function $bb(n)$ is not known for $n \geq 5$. Does Anyone know 
suitable lower bounds for $bb(7)$ and $bb(8)$? 
Remark: $bb(6)$ as a trivial lower bound does not count as a suitable bound.

REPLY [4 votes]: Recently on the Googology Wiki, user  Wythagoras modified the Busy Beaver machine for six states to create a seven state machine that prints more than $10^{10^{10^{10^{18705353}}}}$ ones.  The machine is listed here, and the computation is verified here. So $S(7) > \Sigma(7) > 10^{10^{10^{10^{18705353}}}}$.
I haven't heard of any good bounds for $\Sigma(8)$.