Largest number that can be written using N characters I remember reading in RPF's biographical book, "Surely you're joking Mr. Feynman" that he used to have timed contest about writing down the biggest number using standard symbols. 
Instead of the time restriction, I was wondering what if we restrict the number of characters, to say $N=10$. What is the largest number we can write using $N$ characters using numbers and standard functions known to most mathematicians? Of course I think the real difficulty would be in proving the said number is the largest. What is the largest number you can think of with $N=10$?
 A: Well, just for fun I thought I'd try to answer the question with the operations restricted to:


*

*nullary operations $\{0,1,2,\ldots,9\}$ and

*binary operation $*$ (i.e. multiplication).


It turns out that
9999999999

is the greatest in this case.  The closest runner-up is:
99999*9999 = 999890001

We can make some simplifying assumptions:  (a) all 10 slots are to be used (otherwise, we just add in another 9 at the end) and (b) any of $0,1,2\ldots,8$ is better off being replaced by a $9$.
This exercise didn't seem all that satisfying really, there's so many sequences that result in syntax errors.  E.g.:
*999*99999
999999999*
999999**99

In my opinion, it gets even less satisfying if you keep going.  E.g.


*

*Is 1/0 allowed?

*Does exponentation require a character?  E.g. 2^6 vs. $2^6$.

*Do you require suitable bracketing?  When?  1/2^{-6} vs. 1/2^-6.  What about 9^9^9?

*How many characters does $\log$ require?

*The result varies on human factors -- e.g. what number system is used (base 10?), how many characters it takes to write a function, etc.

*And, of course, there's the inevitable question of self-referential sequences (which might be difficult to arrange for 10 characters).

A: If tetration is allowed: 9↑↑↑↑↑↑↑↑↑9
A: 9!!!!!!!!! would be the largest number, with n! meaning the factorial of n. If you allow for an ellipsis, and allow for extended real numbers, then 9999999... (infinitely repeating 9's) could be considered as the largest.
A: 9^9^9^9^9^9^9^9^9^9 ofcourse you have to start powering from right side... this number is so big that is beyond all humans comprehension. It is more than elementary particles in whole universe...
