Largest Integer Expressible by 3 Digits

Theorem

The largest integer that can be represented using no more than $3$ digits, with no additional symbols, is:

$9^{9^9} = 9^{387 \, 420 \, 489}$

and (at $369 \, 693 \, 100$ digits, is too large to be calculated on a conventional calculator.

Note that this does not include the notation for tetration: ${}^9 9$.

$\map {\log_{10} } {9^{9^9} } \approx 369 \, 693 \,099 \cdotp 63157 \, 03685 \, 87876 \, 1$

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