  We present a comprehensive study of the frequency shifts associated with the
lattice potential for a Sr lattice clock. By comparing two such clocks with a
frequency stability reaching $5\times 10^{-17}$ after a one hour integration
time, and varying the lattice depth up to $U_0=900 \, E_r$ with $E_r$ being the
recoil energy, we evaluate lattice related shifts with an unprecedented
accuracy. We put the first experimental upper bound on the recently predicted
frequency shift due to the magnetic dipole (M1) and electric quadrupole (E2)
interactions. This upper bound is significantly smaller than the theoretical
upper limit. We also give a new upper limit on the effect of
hyperpolarizability with an improvement by more than one order of magnitude.
Finally, we report the first observation of the vector and tensor shifts in a
lattice clock. Combining these measurements, we show that all known lattice
related perturbation will not affect the clock accuracy down to the $10^{-17}$
level, even for very deep lattices, up to $U_0=150\,E_r$.
