Surely something like (n+1)-power-over-n-power should be representable by iterating quad-over-lins. It corresponds to a power cone. Homogenizing an example from https://docs.mosek.com/modeling-cookbook/cqo.html#simple-sets-involving-power-functions we have that xy^2\geq z^3 with x,y,z\geq 0 is equivalent to sy\geq z^2,\ xz\geq s^2.