Curious animal. [$]\sqrt{-2\ln(1-\frac{1}{\sqrt[n]{2}})}[$] would be the median?Thanks.
No, first, I said "n is large but not infinite", so the puzzle is to provide an asymptotic formula that depends upon n. Keep thinking!
Second, if n were infinite, the expected max is [$]+\infty[$].
Hint: start from bearish's comment, generalize it, and start calculating.