  We calculate the photoproduction of double $J/\psi$ ($\Upsilon$) to leading
order based on the nonrelativistic quantum chromodynamics factorization
framework at the Large Hadron Collider with forward proton tagging. The
numerical results of double $J/\psi$ photoproduction pp $\rightarrow$
p$\gamma$p $\rightarrow$ $J/\psi$ + $J/\psi$ with different forward detector
acceptances ($\xi$) are presented. The total cross section of double $J/\psi$
photoproduction is less than 200 fb with 0.1 $<$ $\xi$ $<$ 0.5, but can reach
about 1.37(1.27) pb with 0.0015 $<$ $\xi$ $<$ 0.5 ( 0.0015 $<$ $\xi$ $<$ 0.15
). The double $J/\psi$ photoproduction may have the potential to be detected
and provide an interesting signature, thus is useful for studying the mechanism
of heavy quarkonium production. We also predict the double $\Upsilon$
photoproduction and find they are, unfortunately, small (with less than 10 fb).
