Just as a quick example of why it's a bit absurd - I name that number you just defined $zeta$. Now I make $zeta'$ = zeta^zeta. Or whatever manipulation you like. Adding constraints is addressed in the link.
The GP question was not about encoding, and thus is not subject to compression. The largest number we can measure of anything is a pretty well defined concept.
We don’t know if there is only a finite number of discrete states in the observable universe. The Planck length is not a discretization.
I'll have to think about this. I want a very physical meaning of the term using values rather than references. One where the largest number that can be expressed using up/down fingers with two hands is 1024, not a sign language reference to a googolplex.
"Measuring the intelligence of an idealized mechanical knowing agent" https://philpapers.org/archive/ALEMTI-2.pdf
"Intuitive Ordinal Notations" https://github.com/semitrivial/IONs
https://en.m.wikipedia.org/wiki/Hyperoperation
I wonder if it would be helpful to restrict the question a bit, maybe something like: what is the largest number for which the number, and also all smaller magnitude integers, can be expressed.
All digital representations rely on discrete states in hardware, and there’s a finite number of those in the observable universe, so there should a finite maximum number for computers.