Using arrow-notation - Graham's number

You might be wondering what the previous post on arrow notation can be used for. It is mainly a useless convention that is inapplicable in most contexts, because the numbers involved are so big as demonstrated by the 3↑↑↑↑3 case.

However, it is an integral part in explaining the largest named number in existence. Obviously, there is no such thing as a biggest number, but this one has actual meaning. And, it has a name: Graham's number.
Named after Ronald Graham, the number arose from solving a problem in a branch of mathematics called Ramsey theory that involved coloring graphs on a hypercube (4-dimensional).

Now, how do we calculate Graham's number?
We first start with 3↑↑↑↑3. Calculate this number (which as we found last time, was really really HUGE), and call it "G₁".
Now, find 3↑↑...↑↑3, where there are "G₁" arrows. Let this new giant number be G₂.
Calculate 3↑↑...↑↑3, where there are "G₂" arrows. Let this be G₃.
And so on, until you get to G₆₄=G, Graham's number.
Yes it is very large, larger than anyone can ever imagine, and nothing can come close to express the magnitude of the number in our universe (even 3↑↑↑↑3 was impossible, so G is just beyond impossible!).