Let $(X,Y)$ be a random vector taking values in $\mathbb{R}^2$ such that both $X$ and $Y$ are (marginally) distributed uniformly over the interval [0,1]. Let $Z= \max(X,Y)$ and $f$ denote the density of $Z$.
When $X$ and $Y$ are independent, we have that $f(z)=2z$ on the support of $X$. Hence, the (essential) supremum of the density of Z is 2. My question is:
Can I make the supremum of the density of $Z$ greater than 2 (the independent case) if I consider different couplings of $(X,Y)$ with uniform $(0,1)$ marginals, or is the independent case the worst case?
If so, does this result hold in general, i.e., given any two marginal densities, the supremum is active when they are coupled independently? Can it be extended to any finite number of random variables?