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Based on current CERN infrastructure, an electron-proton collider is proposedat a centre-of-mass energy of about 9 TeV. A 7 TeV LHC bunch is used as theproton driver to create a plasma wakefield which then accelerates electrons to3 TeV, these then colliding with the other 7 TeV LHC proton beam. The basicparameters of the collider are presented, which although of very high energy,has integrated luminosities of the order of 1 pb$^{-1}$/year. For such acollider, with a centre-of-mass energy 30 times greater than HERA, partonmomentum fractions, $x$, down to about $10^{-8}$ are accessible for $Q^2$ of 1GeV$^2$ and could lead to effects of saturation or some other breakdown ofDGLAP being observed. The total photon-proton cross section can be measured upto very high energies and also at different energies as the possibility ofvarying the electron beam energy is assumed; this could have synergy withcosmic-ray physics. Other physics which can be pursued at such a collider arecontact interaction searches, such as quark and electron substructure, andmeasurements of the proton structure as well as other more conventionalmeasurements of QCD at high energies and in a new kinematic regime. The eventsat very low $x$ will lead to electrons and the hadronic final state produced atvery low angles and so a novel spectrometer device will be needed to measurethese. First ideas of the physics programme of such a collider are given.

VHEeP: A very high energy electron-proton collider based on proton-driven plasma wakefield acceleration