Definition:Graham's Number

Definition
Let $3 \uparrow 3$ denote Knuth's notation for powers:
 * $3 \uparrow 3 := 3^3$

Further, let:
 * $3 \uparrow \uparrow 3 := 3 \uparrow \left({3 \uparrow 3}\right) = 3^{\left({3^3}\right)}$

and so define:
 * $3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_n 3 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n - 1} \left({3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n - 1} 3}\right)$

Thus, for example:
 * $3 \uparrow \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow \uparrow \left({3 \uparrow \uparrow \uparrow 3}\right)$

Let:
 * $n_1 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{3 \uparrow \uparrow \uparrow \uparrow 3} 3$

That is, a total of $3 \uparrow \uparrow \uparrow \uparrow 3$ instances of $\uparrow$.

Similarly, let:
 * $n_2 := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_1} 3$

In general for $m \ge 2$:
 * $n_m := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_{m - 1} } 3$

and so specifically:
 * $n_{63} := 3 \underbrace {\uparrow \uparrow \ldots \uparrow \uparrow}_{n_{62} } 3$

It is this $n_{63}$ which is defined as Graham's number.