The Greatest with Three Digits

Using only three digits, and any mathematical signs you like, what is the greatest number you can write? Roughly how many digits would it have?

Solution

The largest is the power tower $$9^{9^{9}},$$ read from the top down: first $9^9$, then $9$ raised to that. It dwarfs the alternatives. Writing the digits side by side gives only $999$; multiplying or taking ordinary powers like $(9^9)^9 = 9^{81}$ stays far smaller; the height of a tower beats them all. Note that $9^{9^9}$ means $9^{(9^9)}$, not $(9^9)^9$, and the difference is enormous: the exponent is $9^9 = 387{,}420{,}489$.

So the number is $9$ raised to nearly $387$ million. Its number of digits is about $$9^9 \times \log_{10} 9 \approx 387{,}420{,}489 \times 0.954 \approx 370 \text{ million}.$$ Printed at a few digits to the centimetre it would run for hundreds of kilometres.