Krylov complexity is an important dynamical quantity with relevance to the study of operator growth and quantum chaos and has recently been much studied for various time-independent systems. We initiate the study of K complexity in time-dependent (driven) quantum systems. For periodic time-dependent (Floquet) systems, we develop a natural method for doing the Krylov construction and then define (state and operator) K complexity for such systems. Focusing on kicked systems, in particular the quantum kicked rotor on a torus, we provide a detailed numerical study of the time dependence of Arnoldi coefficients as well as of the K complexity with the system coupling constant interpolating between the weak and strong coupling regimes. We also study the growth of the Krylov subspace dimension as a function of the system coupling constant.