The AGILE gamma-ray burst GRB 080514B is the first detected to have emission above 30 MeV and an optical afterglow. However, no spectroscopic redshift for this burst is known. We report on our ground-based optical/NIR and millimeter follow-up observations of this event at several observatories, including the multi-channel imager GROND on La Silla, supplemented by Swift UVOT and Swift XRT data. The spectral energy distribution (SED) of the optical/NIR afterglow is found to decline sharply bluewards to the UV bands, which can be utilized in estimating the redshift. Fitting the SED from the Swift UVOT uvw2 band to the H band, we estimate a photometric redshift of z=1.8+0.4-0.3, which is consistent with the reported pseudo-redshift based on gamma-ray data. We find that the afterglow properties of GRB 080514B do not differ from those exhibited by the global sample of long bursts. Compared with the long burst sample, we conclude that this burst was special because of its high-energy emission properties, even though both its afterglow and host galaxy are not remarkable in any way. Obviously, high-energy emission in the gamma-ray band does not automatically correlate with the occurrence of special features in the corresponding afterglow light. Appendix A is only available in electronic form at http://www.aanda.org