The Guin Constant

I was reading James Apnes Notes on Discrete Mathematics  this last week and came across this statement in section 2.1.5:

"The natural numbers N. These are defined using the Peano axioms,
and if all you want to do is count, add, and multiply, you don’t need
much else. (If you want to subtract, things get messy.)"

Since the book is aimed at Computer Science majors at Yale, I was bothered by this statement right off the bat.  Computers are pretty good at some aspects of mathematics, but not all.

In this case, I can't use a computer to count.  Eventually, my computer will run out of memory. When I try to add one more to the current number, my computer will crash.  I know what you are thinking: "I'll just add more memory!"  (It's turtles all the way down).  That won't work either since memory is made out of matter, and the universe in which we live has some finite amount of matter.

So suppose we used each bit of matter (atoms, leptons, whatever smaller building block we discover next) 100% efficiently to store the biggest number we possibly can store in this universe.

I call that the Guin Constant.   It is the biggest number this universe can hold precisely using all matter to store it.

And please don't add one to it.  I'd hate to crash this universe.

Questions, comments, concerns and criticisms always welcome,
John

PS: apologies to mathematicians if this constant already exists.  It's hard to find information like this with internet searches and/or the math books within arms reach of me right now.