Again, you're implicitly assuming the representation has to be a string of digits. I'm not sure that I can convince you, or even what to convince you of, as you aren't accepting the premises of the argument. The numbers +/- 1% of Graham's number can be trivially represented with a program. Likewise, here's some people codegolfing programs to output Graham's number[1]. If it seems like I'm cheating by using too powerful of a method, that's the point that the original person was making: there exist real numbers that can't be described even like this.
[1]: https://codegolf.stackexchange.com/questions/83873/theoretic...