Dynamical Dark Matter (DDM) is an alternative framework for dark-matter physics in which the dark sector comprises a vast ensemble of particle species whose decay widths are balanced against their cosmological abundances. Previous studies of this framework have focused on a particular class of DDM ensembles --- motivated primarily by KK towers in theories with extra dimensions --- in which the density of states scales roughly as a polynomial of mass. In this paper, by contrast, we study the properties of a different class of DDM ensembles in which the density of states grows exponentially with mass. Ensembles with this Hagedorn-like property arise naturally as the "hadrons" associated with the confining phase of a strongly-coupled dark sector; they also arise naturally as the gauge-neutral bulk states of Type I string theories. We study the dynamical properties of such ensembles, and demonstrate that an appropriate DDM-like balancing between decay widths and abundances can emerge naturally --- even with an exponentially rising density of states. We also study the effective equations of state for such ensembles, and investigate some of the model-independent observational constraints on such ensembles that follow directly from these equations of state. In general, we find that such constraints tend to introduce correlations between various properties of these DDM ensembles such as their associated mass scales, lifetimes, and abundance distributions. For example, we find that these constraints allow DDM ensembles with energy scales ranging from the GeV scale all the way to the Planck scale, but the total present-day cosmological abundance of the dark sector must be spread across an increasing number of different states in the ensemble as these energy scales are dialed from the Planck scale down to the GeV scale. Numerous other correlations and constraints are also discussed.

29 pages, LaTeX, 10 figures