The accurate description of nuclear quantum effects, such as zero-point energy, is important for modeling a wide range of chemical and biological processes. Within the nuclear–electronic orbital (NEO) approach, such effects are incorporated in a computationally efficient way by treating electrons and select nuclei, typically protons, quantum mechanically with molecular orbital techniques. Herein, we implement and test a NEO coupled cluster method that explicitly includes the triple electron–electron–proton excitations, where two electrons and one proton are excited simultaneously, using automatic differentiation. Our calculations show that this NEO-CCSDTeep method provides highly accurate proton densities and proton affinities, outperforming any previously studied NEO method. These examples highlight the importance of the triple electron–electron–proton excitations for an accurate description of nuclear quantum effects. Additionally, we also implement and test the second-order approximate coupled cluster with singles and doubles (NEO-CC2) method as well as its scaled-opposite-spin (SOS) versions. The NEO-SOS′-CC2 method, which scales the electron–proton correlation energy as well as the opposite-spin and same-spin components of the electron–electron correlation energy, achieves nearly the same accuracy as the NEO-CCSDTeep method for the properties studied. Because of its low computational cost, this method will enable a wide range of chemical and photochemical applications for large molecular systems. This work sets the stage for a variety of developments and applications within the NEO framework.