  We study the processes e+e- -> (J/psi gamma gamma) gamma and e+e- -> (J/psi
pi- pi+) gamma where the hard photon radiated from an initial e+e- collision
with center-of-mass (CM) energy near 10.58 GeV is detected. In the final state
J/psi gamma gamma we consider J/psi pi0, J/psi eta, chi_{c1} gamma, and
chi_{c2} gamma candidates. The invariant mass of the hadronic final state
defines the effective e+e- CM energy in each event, so these data can be
compared with direct e+e- measurements. We report 90% CL upper limits for the
integrated cross section times branching fractions of the J/psi gamma gamma
channels in the Y(4260) mass region.
