  Heisenberg time evolution under a chaotic many-body Hamiltonian $H$
transforms an initially simple operator into an increasingly complex one, as it
spreads over Hilbert space. Krylov complexity, or `K-complexity', quantifies
this growth with respect to a special basis, generated by $H$ by successive
nested commutators with the operator. In this work we study the evolution of
K-complexity in finite-entropy systems for time scales greater than the
scrambling time $t_s>\log (S)$. We prove rigorous bounds on K-complexity as
well as the associated Lanczos sequence and, using refined parallelized
algorithms, we undertake a detailed numerical study of these quantities in the
SYK$_4$ model, which is maximally chaotic, and compare the results with the
SYK$_2$ model, which is integrable. While the former saturates the bound, the
latter stays exponentially below it. We discuss to what extent this is a
generic feature distinguishing between chaotic vs. integrable systems.
