  We study the production of $J/\psi+c\bar{c}$ in $e^+e^-$ annihilation through
two virtual photons. The cross section is estimated to be 23 fb at
$\sqrt{s}=10.6$ GeV, which is smaller by a factor of six than the calculated
cross section for the same process but through one virtual photon. As a result,
while the annihilation into two photons may be important for certain exclusive
production processes, the big gap between the inclusive production cross
section $\sigma(e^+e^- \to J/\psi + c\bar{c})\simeq$ 0.9pb observed by Belle
and the current nonrelativistic QCD prediction of $\simeq$0.15pb still remains
very puzzling. We find, however, as the center-of-mass energy increases
($\sqrt{s}>20$ GeV) the production through two virtual photons $e^+e^- \to
2\gamma^* \to J/\psi + c\bar{c}$ will prevail over that through one virtual
photon, because in the former process the photon fragmentation into $J/\psi$
and into the charmed quark pair becomes more important at higher energies.
