  We report on the observation of cold collisions between $^6$Li atoms and
Yb$^+$ ions. This combination of species has recently been proposed as the most
suitable for reaching the quantum limit in hybrid atom-ion systems, due to its
large mass ratio. For atoms and ions prepared in the $^2S_{1/2}$ ground state,
the charge transfer and association rate is found to be at least~10$^{3}$ times
smaller than the Langevin collision rate. These results confirm the excellent
prospects of $^6$Li--Yb$^+$ for sympathetic cooling and quantum information
applications. For ions prepared in the excited electronic states $^2P_{1/2}$,
$^2D_{3/2}$ and $^2F_{7/2}$, we find that the reaction rate is dominated by
charge transfer and does not depend on the ionic isotope nor the collision
energy in the range $\sim$~1--120~mK. The low charge transfer rate for ground
state collisions is corroborated by theory, but the $4f$ shell in the Yb$^+$
ion prevents an accurate prediction for the charge transfer rate of the
$^2P_{1/2}$, $^2D_{3/2}$ and $^2F_{7/2}$ states. Using \textit{ab initio}
methods of quantum chemistry we calculate the atom-ion interaction potentials
up to energies of 30$\times 10^3$~cm$^{-1}$, and use these to give qualitative
explanations of the observed rates.
