We point out that, for Dirac neutrinos, in addition to the standard thermal cosmic neutrino background (CνB), there could also exist a nonthermal neutrino background with comparable number density. As the right-handed components are essentially decoupled from the thermal bath of standard model particles, relic neutrinos with a nonthermal distribution may exist until today. The relic density of the nonthermal (nt) background can be constrained by the usual observational bounds on the effective number of massless degrees of freedom Neff and can be as large as nνnt 0.5nγ. In particular, Neff can be larger than 3.046 in the absence of any exotic states. Nonthermal relic neutrinos constitute an irreducible contribution to the detection of the CνB and, hence, may be discovered by future experiments such as PTOLEMY. We also present a scenario of chaotic inflation in which a nonthermal background can naturally be generated by inflationary preheating. The nonthermal relic neutrinos, thus, may constitute a novel window into the very early Universe.