Which algorithms have runtime recurrences like $T(n) = \sqrt{n}\,T(\sqrt{n}) + O(n)$?
Think of an algorithm, which do something linear with an integer list of length $n$, for example computes the maximum. Afterwards, the algorithm divides the list of length $n$ into $\sqrt{n}$ lists of length $\sqrt{n}$ and starts the algorithm for them. The result of the algorithm is for example the product of the computed maximum and the results of the $\sqrt{n}$ lists. For the base case, a list of length $1$, you can return  the value of the only element.

This algorithm has the time complexity, you asked for.