In real world applications is there a concrete benefit when using O(log(log(n)) instead of 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 < 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 :

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 $n < 10^6$ 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 :

$\qquad \displaystyle \forall n < 10^6.\ \frac{\log n}{\log(\log(n))} < 5.26146$