Considering the big-o-notation, there are a variety of algorithms that have the $O(N \log N)$ computational complexity; such algorithms are for example the merge sort, fast fourier transform, etc.
The computational complexity $O(N\log(\log N)))$ is seen much less. What algorithms / sequences are limited by this time complexity?
For example, consider the usual divide-and-conquer algorithms. Those algorithms usually split the original problem of $O(N^2)$ complexity recursively into two halves and solve each halve separately, recuding the needed time to $O(N\log N)$. In order to get the $O(N log(log(N)))$ complexity, how much should the problem size be reduced on each step?