We introduce the Rydberg Composite, a new class of Rydberg matter where a single Rydberg atom is interfaced with a dense environment of neutral ground state atoms. The properties of the Composite depend on both the Rydberg excitation, which provides the gross energetic and spatial scales, and on the distribution of ground state atoms within the volume of the Rydberg wave function, which sculpt the electronic states. The latter range from the "trilobites", for small numbers of scatterers, to delocalized and chaotic eigenstates for disordered scatterer arrays, culminating in the dense scatterer limit in symmetry-dominated wave functions which promise good control in future experiments. We characterize these scenarios with different theoretical methods, enabling us to obtain scaling behavior for the regular spectrum and measures of chaos and delocalization in the disordered regime. Thus, we obtain a systematic description of the Composite states. The 2D monolayer Composite possesses the richest spectrum with an intricate band structure in the limit of homogeneous scatterers.

18 pages, 10 figures