Let $$ f(x)=-\log\log x+\sum_{2\le n\le x}\frac{1}{n\log n}. $$
How can I efficiently compute $$ f(\infty)=:\lim_{x\to\infty}f(x)? $$
Brute force suffices to find 0.7946786454... but I would like several hundred digits.
It seems that I should be able to use numerical integration, since $$ \frac{1}{n\log n}-\log\log n+\log\log(n-1) $$ is smooth (and appears to be monotonic). (In fact, it even has a closed-form integral in li.) Alternately, various forms of series acceleration may apply.
(N.B. I have no real experience with numerical analysis outside an undergraduate class a few years back.)
