Is there a complexity degree that is bigger than $O(n)$ and smaller than $O(n \log n)$?
closed as off-topic by Christian Clason, Geoffrey Irving, Geoff Oxberry Jun 2 '14 at 21:23
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$n \log\log n$ is between $n$ and $n \log n$, and is a relatively common one to find in the wild.
On top of $O(n\log(\log(n)))$, there's also $O(n \log^*(n))$ in which $\log^*$ is the number of times the logarithm function must be applied in order for the result to be less than or equal to 1.
For instance, if you already know an Euclidean minimum spanning tree, the Delaunay triangulation may be discovered in $O(n\log^*(n))$ time.
More extremely, one can look at the inverse Ackermann function $\alpha(n,n)$, which may be found in the analysis of several algorithms of complexity $O(n\alpha(n,n))$. There's a good introduction here.
There are infinitely many, since $O(n(\log n)^\alpha) \subsetneq O(n(\log n)^\beta)$ for any $\alpha<\beta$. So, in particular, $O(n) = O(n(\log n)^0) \subsetneq O(n(\log n)^\alpha) \subsetneq O(n\log n)$ for any $\alpha\in (0,1)$.