Does this infinite series over nested logarithm converge?

Question

Define a sequence $(f_i)_{i\in\mathbb{N}}$ via $f_{i+1}(n)=\log(f_i(n)+1)$ and $f_0(n)=n$. Does the sequence $$\left(\sum\limits_{n=1}^\infty\frac{1}{\prod_{j=0}^i f_j(n)}\right)_{i\in\mathbb{N}_0}$$ converge? If not, are there values for $i$, such that the sum converges? The first three elements $$\sum\limits_{n=1}^\infty\frac{1}{n}$$ $$\sum\limits_{n=1}^\infty\frac{1}{n\log(n+1)}$$ $$\sum\limits_{n=1}^\infty\frac{1}{n\log(n+1)\log(\log(n+1)+1)}$$ for example all diverge.