  We present the first calculation of exclusive double $J/\psi$ production in
hadronic collisions. We analyse in detail the form of the Born-level $gg \to
J/\psi J/\psi$ amplitudes within the non-relativistic quarkonium approximation
and discuss the implications of this for the central exclusive production
channel, within the `Durham' perturbative model. In addition we show that this
direct single parton scattering contribution is expected to be strongly
dominant in the exclusive case. We present predictions for the LHC and show
that the expected cross sections are in reasonable agreement with the LHCb
Run-I measurement of exclusive double $J/\psi$ production, with the measured
invariant mass distribution described well by the theory. Motivated by this
encouraging result we present predictions for observables that may be measured
in LHC Run-II, and estimate the size of the expected cross sections in the
$\psi(2S)$ and $\chi_{c}$ cases.
