Using 980 fb(-1) of data at and around the Upsilon(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 gamma gamma -> gamma psi(2S) is studied from the threshold to 4.2 GeV for the first time. Two structures are seen in the invariant mass distribution of gamma psi(2S): one at M-R1 = 3922.4 +/- 6.5 +/- 2.0 MeV/c(2) with a width of Gamma(R1) = 22 +/- 17 +/- 4 MeV, and another at M-R2 = 4014.3 +/- 4.0 +/- 1.5 MeV/c(2) with a width of Gamma(R2) = 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 chi(c2)(3930), and the local statistical significance is determined to be 3.1 sigma with the systematic uncertainties included. The second matches none of the known charmonium or charmoniumlike states, and its global significance is determined to be 2.8 sigma including the look-elsewhere effect.
The production rates are Gamma B-gamma gamma(R-1 -> gamma psi(2S)) = 9.8 +/- 3.6 +/- 1.3 eV assuming (J(PC), vertical bar lambda vertical bar) = (0(++), 0) or 2.0 +/- 0.7 +/- 0.2 eV with (2(++), 2) for the first structure and Gamma B-gamma gamma(R-2 -> gamma psi(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 lambda is the helicity.