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