Starting with a number of at least $9$ digits, every minute we add the sum of the leftmost $9$ digits of the current number to the number itself. Will we always, at some point, see $9$ consecutive numbers that are all not divisible by $9$?
The sum of the leftmost $9$ digits is at most $9\times 9=81$, so we never add more than $81$ each time. Consequently, when the number is large enough, the leftmost $9$ digits will stay the same for more than $9$ minutes. We can wait until the leftmost $9$ digits is divisible by $9$, to make sure that we have $9$ consecutive numbers that leave the same remainder when divided by $9$. If we can ensure that at some point this remainder is not $0$, we would be done.