  A search is performed for the central exclusive production of pairs of
charmonia produced in proton-proton collisions. Using data corresponding to an
integrated luminosity of $3{\rm\ fb}^{-1}$ collected at centre-of-mass energies
of 7 and 8 TeV, $J/\psi J/\psi$ and $J/\psi\psi(2S)$ pairs are observed, which
have been produced in the absence of any other activity inside the LHCb
acceptance that is sensitive to charged particles in the pseudorapidity ranges
$(-3.5,-1.5)$ and $(1.5,5.0)$. Searches are also performed for pairs of P-wave
charmonia and limits are set on their production. The cross-sections for these
processes, where the dimeson system has a rapidity between 2.0 and 4.5, are
measured to be $$ \begin{array}{rl} \sigma^{J/\psi J/\psi} &= 58\pm10{(\rm
stat)} \pm 6{(\rm syst)} {\rm\ pb} , \\ \sigma^{J/\psi\psi(2S)} &= 63
^{+27}_{-18}{(\rm stat)}\pm 10{(\rm syst)} {\rm\ pb} , \\
\sigma^{\psi(2S)\psi(2S)} &< 237 {\rm\ pb}, \\ \sigma^{\chi_{c0}\chi_{c0}} &<
69 {\rm\ nb}, \\ \sigma^{\chi_{c1}\chi_{c1}} &< 45 {\rm\ pb}, \\
\sigma^{\chi_{c2}\chi_{c2}} &< 141 {\rm\ pb}, \\ \end{array} $$ where the upper
limits are set at the 90% confidence level. The measured $J/\psi J/\psi$ and
$J/\psi\psi(2S)$ cross-sections are consistent with theoretical expectations.
