The reaction γp→pK+K− was studied at Jefferson Lab with photon energies from 1.8 to 3.8 GeV using a tagged photon beam. The goal was to search for a Θ++ pentaquark, a narrow, doubly charged baryon state having strangeness S=+1 and isospin I=1, in the pK+ invariant mass spectrum. No statistically significant evidence of a Θ++ was found. Upper limits on the total and differential cross section for the reaction γp→K−Θ++ were obtained in the mass range from 1.5 to 2.0  GeV/c2, with an upper limit for a narrow resonance with a mass MΘ++=1.54  GeV/c2 of about 0.15 nb, 95% C.L.. This result places a stringent upper limit on the Θ++ width ΓΘ++<0.1  MeV/c2.