In my view it is difficult to come up with an alternative to any of the exotic Lie groups, which are unquestionably quite intricate but are also beautiful because they express the properties of certain geometric spaces using both fundamental algebra (i.e., groups) and geometric structures of their own (i.e., Riemannian geometry). I don't know $E_8$ particularly well, but I still have vivid memories of Robert Bryant's lectures describing the structure of $G_2$.