Logarithmic vs double logarithmic time complexity In real world applications is there a concrete benefit when using $\mathcal{O}(\log(\log(n))$ instead of $\mathcal{O}(\log(n))$ algorithms ?

This is the case when one use for instance van Emde Boas trees instead of more conventional binary search tree implementations. 
But for example, if we take $n &lt; 10^6$ then in the best case the double logarithmic algorithm outperforms the logarithmic one by (approximately) a factor of $5$. And also in general the implementation is more tricky and complex. 

Given that I personally prefer BST over VEB-trees, what do you think ?

One could easily demonstrate that :

$\qquad \displaystyle \forall n &lt; 10^6.\ \frac{\log n}{\log(\log(n))} &lt; 5.26146$
Do not forget that $\log n$ still grows exponentially (in $\log(n)$) faster than $\log(\log n)$!

Indeed, if you look at the quotient of $\log(n)$ and $\log(\log(n))$, there is not much impressive to see:


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But still, you get a factor five to six for sizes up to $100000$. Note that larger sizes are not uncommon in practice, and a speedup by that factor is awesome! It may make the difference between having results after lunch or only tomorrow. Be aware that part of the speedup may be eaten away by higher constants of the tree implementation; you would have to plot (or analyse) $c\cdot \log(n)$ and $d\cdot \log(\log(n))$ with $c,d$ the actual runtime constants to get a real picture.

Additionally, what Dave mentions is important: if the operation sped up thusly is executed, say, linearly often, constant speedups become linear speedups, i.e. you may decrease the leading constant of your entire algorithm! As I said above, that is awesome. Just look at what happens if you run the operation $n$ times:


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Now if that's not worth the trouble I don't know what.