We investigate experimentally and theoretically the behavior of falling disks in a fluid and identify several dynamical regimes as a function of the moment of inertia and Reynolds number: steady falling, periodic oscillating, chaotic, and continuous tumbling. One-dimensional iteration maps of the disk angles at turning points were constructed to explore the evolution of the dynamics, as boundaries between these regimes are crossed. We obtain the first experimental evidence of a progression from fixed-point to chaotic motion via a novel type of intermittency, characterized by a discontinuity in the iteration map.