TITLE: Number of digits of the number of digits of the number of digits of $2014^{2014}$
QUESTION [9 upvotes]: How would you solve that problem : 
What is the number of digits of the number of digits of the number of digits of $2014^{2014}$ ?
(for instance the number of digits of $12345678901234567890$ is $20$, and the numbers of digits of $20$ is $2$, and the numbers of digits of $2$ is $1$, so the number of digits of the number of digits of the number of digits of $12345678901234567890$ is $1$)

REPLY [2 votes]: We know the answer is at least $1$ (You can't have a number with zero digits. If we define zero as having zero digits, then that would require $2014^{2014} = 0$).
Suppose the answer is at least $2$. Then the number of digits of the number of digits of $2014^{2014}$ is at least $10$. So the number of digits of $2014^{2014}$ is at least $1,000,000,000$. But, $2014^{2014} < 10000^{2014}$, the latter of which has only $8053$ digits, a contradiction.
Thus the answer is $1$.