Raising Google's Number to itself doesn't make it appreciably bigger. Instead you can pick TREE(3), or SCG(13), or Loader's Number which is about the largest famous one we know how to compute.
Beyond that there's the likes of Busy Beaver numbers and beyond that is Rayo's number.
Btw, Graham's Number is less than BB(49 bits) [1].
https://research.phys.cmu.edu/biophysics/2021/01/09/nobody-c...
I'm sure it can be entertaining to try and deal the largest possible finite amount of damage, which requires finding uncommonly large numbers.