Graham's Number versus another large number I recently read this article about the most damage you can do in a single turn in Magic the Gathering. According to the current version of the deck, that damage is about
a) $2 \rightarrow 17 \rightarrow 417$
using Conway chained arrow notation.
This got me thinking about whether this number is larger than Graham's Number (sorry, not allowed to put more than 2 links in a post). I've tried to work it out some, but I'm having trouble understanding the chained arrow notation, let alone comparing it to Graham's number.
If it helps, I noticed that Graham's number is between
b) $3 \rightarrow 3 \rightarrow 64 \rightarrow 2$
and
c) $3 \rightarrow 3 \rightarrow 65 \rightarrow 2$
But I don't know how that compares to
a) $2 \rightarrow 17 \rightarrow 417$
Thank you for your time, and any answers.
Edit: corrected typo in damage estimate "copied" from MTG site.
 A: Intuitively, Grahams number should be much larger. Let $M=2 \rightarrow 17 \rightarrow 417$; in Knuth's up-arrow notation we have $M=2 \uparrow^{417} 17$. Let $G$ represent Graham's number. That is, $\left.
 \begin{matrix}
  G &=&3\underbrace{\uparrow \uparrow \cdots\cdots\cdots\cdots\cdots \uparrow}3 \\
    & &3\underbrace{\uparrow \uparrow \cdots\cdots\cdots\cdots \uparrow}3 \\
    & &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\
    & &3\underbrace{\uparrow \uparrow \cdots\cdot\cdot \uparrow}3 \\
    & &3\uparrow \uparrow \uparrow \uparrow3
 \end{matrix}
\right \} \text{64 layers}$
This representation is both unwieldy and, I fear, not enlightening. Informally, we can think of $G$ as $3 \uparrow^{\text{massive}}3$. The heuristic argument in comparing two numbers in Knuth notation is that the one with more arrows is larger. This is not always the case (as explored here and briefly here). For us, the number of arrows in $G$ ridiculously overwhelms the number of arrows in $M$ (the former I could never feasibly write out in decimal form and the latter is $3$-digits long). Hence, I think it's fair to say $M < G$. 
A: In case you wonder about an actual proof: $$2 \uparrow^{417} 17 < 3 \uparrow^{417} 17 < 3 \uparrow^{417} 27 < 3 \uparrow^{417} 3 \uparrow^{417} 3 = 3 \uparrow^{418} 3 <  3 \uparrow^{3 \uparrow\uparrow\uparrow\uparrow 3} \; \; 3$$
The fact that Knuth's arrow notation is monotonous in all arguments is proven in
 Saibian, Sbiis. A theorem for Knuth arrows.
