  The hexadecapole deformation ($\beta_4$) of the $^{238}$U nucleus has not
been determined because its effect is overwhelmed by those from the nucleus'
large quadrupole deformation ($\beta_2$) in nuclear electric transition
measurements. In this Letter, we identify the nonlinear response of the
hexadecapole anisotropy to ellipticity in relativistic $^{238}$U+ $^{238}$U
collisions that is solely sensitive to $\beta_4$ and insensitive to $\beta_2$.
We demonstrate this by state-of-the-art hydrodynamic calculations and discuss
the prospects of discovering the $\beta_4$ of $^{238}$U in heavy-ion data at
the Relativistic Heavy Ion Collider.
