  The process $e^+e^-\to\eta\eta\gamma$ is studied in the center-of-mass energy
range 1.17-2.00 GeV using data with an integrated luminosity of 201
pb$^{-1}$collected by the SND detector at the VEPP-2000 $e^+e^-$ collider. The
$e^+e^-\to\eta\eta\gamma$ cross section is measured for the first time. It is
shown that the dominant mechanism of this reaction is the transition through
the $\phi\eta$ intermediate state. Our result on the $e^+e^-\to\eta\eta\gamma$
cross section is consistent with the $e^+e^-\to\phi\eta$ measurement in the
$\phi\to K^+ K^-$ mode. The search for radiative processes contributing to the
$e^+e^-\to\eta\eta\gamma$ cross section is performed, and no significant signal
is observed.
