We present a generalized Floquet formulation of time-dependent current-density-functional theory for nonperturbative treatment of multiphoton processes of many-electron systems in intense monochromatic electric and magnetic fields. It is shown that the time-dependent Kohn-Sham-like equation can be transformed into an equivalent time-independent Floquet Hamiltonian matrix eigenvalue problem. A procedure is outlined for the determination of the dynamical properties in bound-bound multiphoton transitions. For the case of bound-free transitions, such as the multiphoton ionization processes, we introduce the notion of complex density and complex current and present a non-Hermitian Floquet formalism for the calculation of the complex quasienergies of individual spin orbitals and total many-electron systems.