I wonder whether there’s an upper bound on the largest number that can be expressed in the observable universe.

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.

Depends on what you mean by “expressed”. E.g. you can represent a busy beaver that expresses the length of its output.

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.

> Depends on what you mean by “expressed”

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.

If you are OK with a sum constants times powers of two, would you be ok with other hyperoperations on two?

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.

You might enjoy:

"Measuring the intelligence of an idealized mechanical knowing agent" https://philpapers.org/archive/ALEMTI-2.pdf

"Intuitive Ordinal Notations" https://github.com/semitrivial/IONs

all expression is reference

Related to https://en.m.wikipedia.org/wiki/Berry_paradox

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.

And zeta' can not be expressed by any state of the visible Universe.

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.

Tbf though, I'm sure the number of cat pics floating about the internet dwarfs this number, so it depends on the data I suppose.

Plus rather simple things like pi could create rather a long message.

I suppose pi has a cheat code: if you made the largest possible circle in the universe you only need enough digits to distinguish points on its circumference that are a Planck distance apart. Then you can either ignore any digits beyond that or even simply make them up, as there would be no way to measure the difference.