  We present N-body and hydrodynamical simulations of the response of the Milky
Way's baryonic disc to the presence of the Large Magellanic Cloud during a
first infall scenario. For a fiducial galactic model reproducing the gross
properties of the Galaxy, we explore a set of six initial conditions for the
LMC of varying mass which all evolve to fit the measured constraints on its
current position and velocity with respect to the Galactic Center. We find that
the LMC can produce strong disturbances - warping of the stellar and gaseous
discs - in the Galaxy, without violating constraints from the phase-space
distribution of stars in the Solar Neighbourhood. All models correctly
reproduce the phases of the warp and its anti-symmetrical shape about the
disc's mid-plane. If the warp is due to the LMC alone, then the largest mass
model is favoured ($2.5\times10^{11}\,\rm{M_{\odot}}$). Still, some
quantitative discrepancies remain, including deficits in height of $\Delta
Z=0.7 \, \rm{kpc}$ at $R=22 \,\rm{kpc}$ and $\Delta Z=0.7\, \rm{kpc}$ at $R=16
\,\rm{kpc}$. This suggests that even higher infall masses for the LMC's halo
are allowed by the data. A comparison with the vertical perturbations induced
by a heavy Sagittarius dSph model ($10^{11}\,\rm{M_{\odot}}$) suggest that
positive interference with the LMC warp is expected at $R=16 \, \rm{kpc}$. We
conclude that the vertical structure of the Galactic disc beyond the Solar
Neighbourhood may jointly be shaped by its most massive satellites. As such,
the current structure of the MW suggests we are seeing the process of disc
heating by satellite interactions in action.
