Usually, this involves additive or multiplicative persistence of a non-negative integer, which is how often one has to replace the number by the sum or product of its digits until one reaches a single digit.
Because the numbers are broken down into their digits, the additive or multiplicative persistence depends on the radix.
The single-digit final state reached in the process of calculating an integer's additive persistence is its digital root.
The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39 → 27 → 14 → 4.
For any fixed base, the sum of the digits of a number is at most proportional to its logarithm; therefore, the additive persistence is at most proportional to the iterated logarithm, and the smallest number of a given additive persistence grows tetrationally.
For example, the function which takes the minimal digit only allows for persistence 0 or 1, as you either start with or step to a single-digit number.