The sensitivities to anomalous quartic photon couplings at the Large Hadron Collider are estimated using diphoton production via photon fusion. The tagging of the protons proves to be a very powerful tool to suppress the background and unprecedented sensitivities down to $6 cdot 10^{-15}$gev$^{-4}$ are obtained, providing a new window on extra dimensions and strongly-interacting composite states in the multi-TeV range. Generic contributions to quartic photon couplings from charged and neutral particles with arbitrary spin are also presented.
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