Which is bigger: $9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$ or $9!!!!!!!!!$? In my classes I sometimes have a contest concerning who can write the largest number in ten symbols. It almost never comes up, but I'm torn between two "best" answers: a stack of ten 9's (exponents) or a 9 followed by nine factorial symbols. Both are undoubtedly huge, but I haven't been able to produce an argument that one is larger (surely they aren't equal). Any insight into which of these two numbers is bigger would be greatly appreciated.

I find it quite sad that none of the answers has actually formally shown that 

$$9^{9^{9^{9^{9^{9^{9^{9^{9^9}}}}}}}} > 9!!!!!!!!!$$

The proces is a bit long, but it it worth it in my opinion. First, a simple calculation shows $9! < 9^8$. Then $9!! < (9^8)^{9^8} = 9^{8\cdot9^8}$. Then $$9!!!< \left(9^{8\cdot9^8}\right)^{9^{8\cdot9^8}} < \left(9^{9^9}\right)^{9^{8\cdot9^8}}=9^{9^{8\cdot9^8}\cdot9^9}=9^{9^{8\cdot9^8+9}}$$
$$9!!!! < \left(9^{9^{9^9}}\right)^{9^{9^{8\cdot9^8+9}}}=9^{9^{9^{8\cdot9^8+9}}\cdot 9^{9^9}}=9^{9^{9^{8\cdot9^8+9}+9^9}}<9^{9^{9^{8\cdot9^8+10}}}$$
Now, some notation. Let us use $f(x)$ for $9^x$ and $f^k(x)=f(f(... f(f(x))...))$. Let $x!^k$ denote $x!!!...!!!$ with $k$ factorials. 
Clearly, $f(x)+1<f(x+1)$. By induction, $f^t(x)+1<f^t(x+1)$ for any $t$. 
Now we proceed by induction to show that $9!^k < f^{k-1}(8\cdot9^8+k+6)$. The base case has been given above. Assume $4 \leq k \leq 9^8-7$. 
$$9!^{k+1} < \left(f^{k-1}(8\cdot9^8+k+6)\right)^{f^{k-1}(8\cdot9^8+k+6)}=
9^{f^{k-2}(8\cdot9^8+k+6)f^{k-1}(8\cdot9^8+k+6)}<9^{9^{f^{k-3}(9^9)+f^{k-2}(8\cdot9^8+k+6)}}<9^{9^{9\cdot f^{k-2}(8\cdot9^8+k+6)}}<9^{9^{9^{f^{k-3}(8\cdot9^8+k+6)+1}}}<9^{9^{9^{f^{k-3}(8\cdot9^8+k+7)}}}=f^k(8\cdot9^8+k+7)$$
This completes the induction hypothesis. Hence we have shown the asked.

Now for some fun: The largest number using ten symbols. We can also use shortened Knuth's up arrow notation, $9 \uparrow^{9 \uparrow^{9 \uparrow^{9} 9} 9} 9$ has ten symbols.
Conway chained arrow notation, $9 \rightarrow 9 \rightarrow 9 \rightarrow 9 \rightarrow 9^9$.
Rado's sigma function, $\Sigma(\Sigma(\Sigma(9)))$. 
