Largest Integer Expressible by 3 Digits/Number of Digits

Theorem

 * $9^{9^9}$ has $369 \, 693 \, 100$ digits when expressed in decimal notation.

Proof
Let $n$ be the number of digits in $9^{9^9}$

From Number of Digits in Number:
 * $n = 1 + \left \lfloor{\log_{10} \left({9^{9^9} }\right)}\right\rfloor$

where $\left \lfloor{\ldots}\right\rfloor$ denotes the floor function.

Then: