Are there any divergent series that approximate to O(log log log n)?

Given any sequence $b_n$ of positive numbers increasing to $\infty$ there is always a series $\sum a_n$ of positive terms such that $a_1+a_2+\cdots+a_n=b_n$ for all $n$. [Take $a_1=b_1$ and $a_n=b_n-b_{n-1}$ for $n \geq 2$]

Yes. Since $e^{e^e} \approx 3.8\cdot 10^6$ we would like to have our sum $1$ after, say, $3.9\cdot 10^6$ terms, so we start with $3.9 \cdot 10^6$ terms of $\frac 1{3.9\cdot 10^6}$. Then for each successive term $a_n$, take $\frac 1k$ with $k$ the smallest number possible so that the sum does not exceed $\log \log \log n$. Roughly speaking you want $a_n$ to be about $\frac 1{n \log n \log \log n}$. You can continue the pattern with more logs as far as you want.