  Weak lensing will play an important role in future cosmological surveys,
including e.g. Euclid and SKA. Sufficiently accurate theoretical predictions
are important for correctly interpreting these surveys and hence for extracting
correct cosmological parameter estimations. We quantify for the first time in a
relativistic setting how many post-Born and lens-lens coupling corrections are
required for sub-percent accuracy of the theoretical weak lensing convergence
for $z\le 2$ (the primary weak lensing range for Euclid and SKA). We do this by
ray-tracing through a fully relativistic exact solution of the Einstein Field
Equations which consists of randomly packed mass-compensated underdensities of
realistic amplitudes. We find that including lens-lens coupling terms and
post-Born corrections up to second and third order respectively is sufficient
for sub-percent accuracy of the convergence along $94\%$ of the studied light
rays. We also find that a significant percentage of the studied rays have
post-Born corrections of size over $10\%$ of the usual gravitational
convergence, $\kappa^{(1)}$, and several rays even have post-Born corrections
several times the size of $\kappa^{(1)}$ at $z = 2$.
