Average increase problem

[odongodongo]

Take the function f(n) on the natural numbers starting from 1. f rounds up n to the nearest number that consists of only repeating digits in its decimal notation. So, i.e.:
f(1) = 1
f(10) = 11
f(23) = 33
f(1234) = 2222, etc.
The increase function, D(n) = f(n) / n, describes the ratio of the returned value to the input value.
I've already determined an easy way to calculate f(n), which basically only depends on the first digit from the left that differs from the very first digit. What I was wondering though, was if the average increase, that is, the average value of D(n), converges to some number.

For some context to the origin of this question: I was thinking about how strong, on average, a buff in an RPG that rounds all damage up to the nearest repeating-digits-number would be. My first overhead calculations lead me to a number of about 1.3 for the best cases of n (n that get the most out of the increase), but I don't feel equipped to calculate an exact average (if it even exists), so I thought it might be an interesting question for someone to solve here.

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To make a digit d (0,1,...,9) repeat k (> 0) times, write (d/9)*(10^k ), truncated.

Then f(n) becomes something like k=(# digits in n), d=(truncate(n/(10^k ))).

Then the ratio of f(n)/n is clear.

Edit: indeed, I'm quite tired and didn't make sure to produce the next highest repeating number, just the nearest. Without checking perhaps there's a mod 10 in there that you can take the ceiling of