  In the presence of time reversal symmetry, a non-linear Hall effect can occur
in systems without an inversion symmetry. One of the prominent candidates for
detection of such Hall signals are Weyl semimetals. In this article, we
investigate the Berry curvature induced second and third order Hall effect in
multi-Weyl semimetals with topological charges $n =1, 2, 3$. We use low energy
effective models to obtain general analytical expressions and discover the
presence of a large Berry curvature dipole in multi-Weyl semimetals. We also
study the Berry curvature dipole in a realistic tight-binding lattice model and
observe two different kinds of variation with increasing topological charge --
these can be attributed to different underlying Berry curvature components. We
provide estimates of the signatures of second harmonic of Hall signal in
multi-Weyl semimetals, which can be detected experimentally. Furthermore, we
predict the existence of a third order Hall signal in multi-Weyl semimetals. We
derive the analytical expressions of Berry connection polarizability tensor,
which is responsible for third order effects, using a low energy model and
estimate the measurable conductivity. Our work can help guide experimental
discovery of Berry curvature multipole physics in multi-Weyl semimetals.
