By integral comparison it is fairly easy to prove that the series $$ \sum_{n\geq N_k}\frac{1}{\prod_{i=0}^k\log^{(i)}(n)} $$ diverges for all integers $k\geq 0$, where for an integer $i\geq 1$ $$ \log^{(i)}(n):=\underbrace{\log(...\log(}_{i\text{ times}}n)...) $$ and $\log^{(0)}(n):=n$, and where $N_k$ is sufficiently large such that all terms of the sum are properly defined.
Now I wonder: if we define $$ f(n):=\max\{i\in\mathbb{N}_0:\log^{(i)}(n)> 1\}, $$ then does the series $$ \sum_{n\geq 3}\frac{1}{\prod_{i=0}^{f(n)}\log^{(i)}(n)} $$ converge? And if so, how about the bigger series $$ \sum_{n\geq 3}\frac{1}{\prod_{i=0}^{f(n)+1}\log^{(i)}(n)} $$ ?
Edit: and what about the series $$ \sum_{n\geq 3}\frac{1}{\prod_{i=0}^{f(n)+2}\log^{(i)}(n)}; $$ is it conditionally convergent, or perhaps even absolutely convergent?