I don’t think Graham’s number in particular has any real-world application.
Graham’s number is from a field called Ramsey theory, which basically asks “how big does X need to be to guarantee that condition Y is met”?
Ramsey theory does have some real-world applications when it comes to graph problems. Graph problems are everywhere in computer science and engineering.
Also, if you’re interested in large numbers (that dwarf Graham’s number), I suggest reading this essay: [Who can name the bigger number?](https://www.scottaaronson.com/writings/bignumbers.pdf).
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