Atomically precise epitaxial structures are unique systems for tunneling spectroscopy that minimize extrinsic effects of disorder. We present a systematic tunneling spectroscopy study, over a broad doping, temperature, and bias range, in epitaxial c-axis La2−xSrxCuO4/La2CuO4/La2−xSrxCuO4 heterostructures. The behavior of these superconductor/insulator/superconductor (SIS) devices is unusual. Down to 20 mK there is complete suppression of c-axis Josephson critical current with a barrier of only 2 nm of La2CuO4, and the zero-bias conductance remains at 20–30% of the normal-state conductance, implying a substantial population of in-gap states. Tunneling spectra show greatly suppressed coherence peaks. As the temperature is raised, the superconducting gap fills in rather than closing at Tc. For all doping levels, the spectra show an inelastic tunneling feature at ∼80 meV, suppressed as T exceeds Tc. These nominally simple epitaxial cuprate junctions deviate markedly from expectations based on the standard Bardeen-Cooper-Schrieffer theory.