Multi-Weyl semimetals are variations of Weyl semimetals characterized by isolated band touching points, each carrying multiple topological charges. Given a plethora of exotic transport properties arising in such systems, it remains a longstanding interest to explore other variations of these semimetal phases. Of particular significance are hybrid multi-Weyl semimetal phases where various isolated band touching points, the number of which can be increased on-demand by tuning system parameters, carrying different topological charges coexist in the same setting. The experimental realization of such systems is expected to allow, in principle, clearer and more distinguishable signatures of isolated band touching points with various topological charges. In this work, we attempt to theoretically devise such systems by means of Floquet engineering. Specifically, we present three separate periodically driven systems displaying single-Weyl, double-single-Weyl, and triple-single-Weyl semimetal phases, each of which is capable of hosting a large number of isolated band touching points. We further report their intricate Fermi arc structures that result from the interplay between isolated band touching points of different charges. Moreover, we characterize these multi-Weyl nodes by use of a dynamical winding invariant.