Using 980 fb-1 of data at and around the I (nS) (n=1, 2, 3, 4, 5) resonances collected with the Belle detector at the KEKB asymmetric-energy e+e-collider, the two-photon process γγ→γψ(2S) is studied from the threshold to 4.2 GeV for the first time. Two structures are seen in the invariant mass distribution of γψ(2S): one at MR1=3922.4±6.5±2.0 MeV/c2 with a width of ΓR1=22±17±4 MeV, and another at MR2=4014.3±4.0±1.5 MeV/c2 with a width of ΓR2=4±11±6 MeV; the signals are parametrized with the incoherent sum of two Breit-Wigner functions. The first structure is consistent with the X(3915) or the χc2(3930), and the local statistical significance is determined to be 3.1σ with the systematic uncertainties included. The second matches none of the known charmonium or charmoniumlike states, and its global significance is determined to be 2.8σ including the look-elsewhere effect. The production rates are ΓγγB(R1→γψ(2S))=9.8±3.6±1.3 eV assuming (JPC,|λ|)=(0++,0) or 2.0±0.7±0.2 eV with (2++,2) for the first structure and ΓγγB(R2→γψ(2S))=6.2±2.2±0.8 eV with (0++,0) or 1.2±0.4±0.2 eV with (2++,2) for the second. Here, the first errors are statistical and the second systematic, and λ is the helicity.