Looking for distributions for which distribution of maximum has a known distribution where the base distributions has continuous positive supports.

Question

Suppose $X$ and $Y$ are two positive continuous distributions and they are independent. Let $Z = \max(X,Y)$. The distribution of $Z$ can be calculated using the probability $$P(X \le z, Y \le z)=P(X \le z)P(Y \le z).$$ But I'm looking for $X$ and $Y$ distributions for which $Z$ has a known probability distribution function. As an example if both $X$ and $Y$ are Frechet distribution with cumulative distribution function $$F(t)=\exp(-\lambda t^{-\alpha}).$$ Then $Z$ is also a Frechet variable. I am looking for similar examples. Not necessary that $X$ and $Y$ have same distributions, but they have to be positive, and $Z$ should have any known distribution. Also if $X$ and $Y$ are same, one can think of a particular distribution for which largest order statistic has some known distribution.