Observations from the Solar Isotope Spectrometer and the Ultra Low Energy Isotope Spectrometer on the ACE spacecraft during the extremely large events of October and November 2003 are combined to create heavy ion spectra over more than 3 decades in energy. The resulting spectra differed substantially in shape from event to event, as well as from element to element within a given event, resulting in energy-dependent abundance ratios. Although the effects of strong local shock acceleration are apparent in the intensities of the 28 and 29 October events, these do not explain the order of magnitude differences between the event-integrated abundances obtained at 0.64--0.91 MeV/nucleon and those at 12--60 MeV/nucleon. The higher-energy abundances relative to the lower-energy ones show trends with nuclear charge or charge-to-mass ratio that are similar for all the events and suggest that heavier ions are less efficiently accelerated to high energies. The position of the breaks in the energy spectra of O, Ne, Mg, Si, S, Ca, and Fe can be understood in terms of leakage from the shock region, if the mean free path is assumed to be a power law in rigidity. The resulting rigidity dependence is consistent with a source of wave turbulence in the vicinity of the shock when the ions are accelerated.