The main idea here is to show there are very few possibilities. It's clear (in a variety of ways) that the number of digits in $n^n$ grows faster than $n$. Therefore, to solve your problem, all you have to do is to keep checking $1, 2, 3, \cdots$ until you get to the point where it's clear $n$ can no longer catch up to the number of digits in $n^n$.
At $n=10$, $10^{10}$ has 11 digits. Past this point, every time you increase $n$ by 1, you add at least one more digit to $n^n$, so this is as far as you need to check.
Really, the main difficulty here is not to overthink the problem; it's easy to overlook the easy solutions if you're too busy looking for hard ones.