User blog:Spitemaster/How big is this function based on the Harmonic series?

First, js code implementing this function:

The function takes the sum of 1/(n * ln(n) * ln(ln(n)) * ... * ln^k(n)) until the value is k. (Well, the way it's implemented, it counts down, but...). It returns the value of n when it reaches there. To keep it strictly positive and real-valued, it sets ln(x) to 1 if it would otherwise be less than 1.

This is large because it is effectively computing the eventually slowest-growing strictly positive real-valued divergent series. Now, there isn't a slowest-growing divergent series, as you can always add more ln(...ln(n)..) on to the end. But you can't do better than this sequence - any slower and it converges. In particular, if S=1/(n * ln(n) * ln(ln(n)) * ... * (ln^k(n))^a) and a>1, the sum of the sequence converges.

By counting how long it takes to converge, we get a large number. But I'm uncertain as to how big. https://math.stackexchange.com/questions/452053/is-there-a-slowest-rate-of-divergence-of-a-series says that 1/(n * ln(n) * ln(ln(n)) * ln(ln(ln(n)))) requires n > googolplex before it reaches 10, so g(10) is obviously much larger.

I wouldn't be surprised if it's slow initially (compared to functions usually found here). I would be surprised if it's not eventually fast. Any help?