We investigate a family of multiple-stable processes that may exhibit either long-range or short-range dependence, depending on the parameters. There are two parameters for the processes, the memory parameter $β\in(0,1)$ and the multiplicity parameter $p\in\mathbb N$. We investigate the macroscopic limit of extremes of the process, in terms of convergence of random sup-measures, for the full range of parameters. Our results show that (i) the extremes of the process exhibit long-range dependence when $β_p := pβ-p+1\in(0,1)$, with a new family of random sup-measures arising in the limit, (ii) the extremes are of short-range dependence when $β_p<0$, with independently scattered random sup-measures arising in the limit, and (iii) there is a delicate phase transition at the critical regime $β_p = 0$.