Pages $1$ to $9$ inclusive require $9$ digits; pages $10$ to $99$ inclusive require $180$ digits. So, in total, $189$ digits are required to number all of the pages before the three-digit page numbers commence with page number $100$.
This leaves $663$ digits, so the last page in the book is the $221^\mbox{st}$ page which has a three-digit number, namely page $320$.