Optical and infrared spectroscopy are presented for the classical nova V3666 Ophiuchi (Nova Ophiuchi 2018 No.3) from 11 months after its outburst. At that epoch, V3666 Oph displayed an [Fe X] λ0.6375 emission line stronger than Hα and nearly a factor of 3 larger, relative to Hα, than in any nova observed previously. The [Fe X] feature, with an equivalent width exceeding 1300 Å, was enormously strong relative to the continuum emission as well. V3666 Oph also reached a state of high excitation for an Fe II type nova, displaying coronal lines of [Fe XI], [S IX], and [Ar X], and permitted features that can be identified with the C VI (8-7), (9-8), and (10-9) transitions. In addition to [Fe X] and [Fe XI], V3666 Oph also displayed bright lines of [Fe VII], indicating that the prominent iron features were not confined to a single or narrow range of ionization states. Modeling the emission line spectrum of V3666 Oph with the photoionization code Cloudy indicated that the bulk of the emission line gas was matter-bounded, severely reducing the amount of low ionization gas and resulting in coronal lines that were very strong in comparison to lines such as H I and He I, and the familiar low excitation forbidden lines of nitrogen and oxygen. Such matter-bounded regions are observationally rare in novae since they are both faint and short-lived, offering a plausible explanation for why [Fe X] λ0.6375 had not be seen at this strength previously.