I don't know if this is the sort of thing you are looking for, but the higher rational homotopy groups appear naturally in representation theory. For example, if $G$ is a simply connected compact Lie group, then the algebra
$Ext^\ast_{\mathcal U (\mathfrak g)}(\mathbb C, \mathbb C) = H^\ast(\mathfrak g)$
is a model for the cochain algebra, so it encodes the rational homotopy groups by rational homotopy theory. Using this, one can show that such group is rationally homotopic to a product of odd spheres.