Graham's number

Graham's number is one of the biggest numbers ever used in a constructive/practical way. It uses Knuth uparrow notation.

Definition

Graham's number uses the G function, with $G_{64}$ being Graham's number.

$G_1$ would be $3 \uparrow \uparrow \uparrow \uparrow 3$. This number alone is ridiculously big, and is pretty much unimaginable. However, $G_2$ would be $3 \uparrow \cdots G_1 \cdots \uparrow 3$. $G_3$ would be $3 \uparrow \cdots G_2 \cdots \uparrow 3$, and so on until $3 \uparrow \cdots G_{63} \cdots \uparrow 3$, which is $G_{64}$.

Graham's number can also be represented like this:

$\left. \begin{matrix} G &=&3\underbrace{\uparrow \uparrow \cdots \cdots \cdots \cdots \cdots \uparrow}3 \\ & &3\underbrace{\uparrow \uparrow \cdots \cdots \cdots \cdots \uparrow}3 \\ & & \underbrace{\qquad \quad \vdots \qquad \quad} \\ & &3\underbrace{\uparrow \uparrow \cdots \cdots \uparrow}3 \\ & &3\uparrow \uparrow \uparrow \uparrow3 \end{matrix} \right \} \text{64 layers}$