We present high-resolution near-infrared spectra taken during eight transits of 55 Cancri e, a nearby low-density super-Earth with a short orbital period (<18 hr). While this exoplanet's bulk density indicates a possible atmosphere, one has not been detected definitively. Our analysis relies on the Doppler cross-correlation technique, which takes advantage of the high spectral resolution and broad wavelength coverage of our data, to search for the thousands of absorption features from hydrogen-, carbon-, and nitrogen-rich molecular species in the planetary atmosphere. Although we are unable to detect an atmosphere around 55 Cancri e, we do place strong constraints on the levels of HCN, NH3, and C2H2 that may be present. In particular, at a mean molecular weight of 5 amu, we can rule out the presence of HCN in the atmosphere down to a volume mixing ratio (VMR) of 0.02%, NH3 down to a VMR of 0.08%, and C2H2 down to a VMR of 1.0%. If the mean molecular weight is relaxed to 2 amu, we can rule out the presence of HCN, NH3, and C2H2 down to VMRs of 0.001%, 0.0025%, and 0.08%, respectively. Our results reduce the parameter space of possible atmospheres consistent with the analysis of Hubble Space Telescope/WFC3 observations by Tsiaras et al. and indicate that if 55 Cancri e harbors an atmosphere, it must have a high mean molecular weight or clouds.