  In the spirit of Mueller-Navelet dijet production, we propose and study the
inclusive production of a forward $J/\psi$ and a very backward jet at the LHC
as an observable to reveal high-energy resummation effects \`a la BFKL. We
obtain several predictions, which are based on the various mechanisms discussed
in the literature to describe the production of the $J/\psi$, namely, NRQCD
singlet and octet contributions, and the color evaporation model.
