We will say that the measure of a number is equal to the maximum degree in which it is possible to represent a number in the form of a sum of digits copied (You can not rearrange the numbers). For example, for $55$ this will be $5$, because $$ 55^1 = 55, \quad 55 = 55$$ $$ 55^2 = 3025, \quad 30+25 = 55 $$ $$ 55^3 = 166375, \quad 1+6+6+37+5 = 55$$ $$ 55^4 = 9150625, \quad 9+15+0+6+25 = 55 $$$$55 ^ 5 = 503284375, \quad 5 + 0 + 3 + 28 + 4 + 3 + 7 + 5 = 55.$$ Let $a_n$ be a sequence of numbers such that all smaller ones have a measure less than. What is the asymptotics of this sequence? Is it possible to somehow build numbers with a given measure? If not, what measures can not be built?
The task was put in a Russian forum, I put it a little different question, I will be glad to any help in its solution :)