In five and higher dimensions, there are $3$ regular polytopes, the hypercube, simplex and cross-polytope. [...] Most uniform higher-dimensional polytopes are obtained by modifying the regular polytopes, or by taking the Cartesian product of polytopes of lower dimensions.
In six, seven and eight dimensions, the exceptional simple Lie groups, $E_6$, $E_7$ and $E_8$ come into play.[...]
I am interested in a quantification of the emphasized sentence above. In what sense are these most uniform polytopes? Does the world of uniform polytopes become "boring" in, say, more than $30$ dimensions, because there only remain simple variations on regular polytopes and cartesian products? Or are there exceptional polytopes expected in higher dimensions too?
I would also be satisfied with something like "this seems to be unknown", preferably with some reference.