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.
EDIT: I thought maybe I should add some extra detail.
The cohomology algebra $H^\ast(G; \mathbb C) = H^\ast(\mathfrak g; \mathbb C)$ is a graded commutative Hopf algebra, and by a theorem of Hopf must be a symmetric algebra $S(V)$ on a graded vector space $V$ in odd degrees ("symmetric algebra" here is meant in the graded sense; as an ungraded algebra it looks like an exterior algebra). Sullivan's rational homotopy theory says that
$\pi_n(G) \otimes \mathbb C = V^\ast$.
Moreover, the vector space $V$ can be explicitly computed as follows. The cohomology of the classifying space $H^\ast(BG)$ is given by the $W$ invariants in $H^\ast(BT)$, where $T$ is a maximal torus and $W=N_G(T)/T$ is the Weyl group. The classifying space $BT$ is a product of $\mathbb CP^\infty$'s, thus $H^\ast(BT)$ is a polynomial algebra on the vector space $\mathfrak t^\ast$ in degree $2$. By a theorem of Chevalley, $H^\ast(BT)^W$ is also polynomial algebra on some graded vector space in even degrees. This vector space precisely $V^\ast[-1]$.
In particular, this shows that $\pi_2(G)$ is torsion. On the other hand, consider the flag variety $G/T$. This is a compact complex manifold with a complex cell structure. Thus its cohomology is only in even degrees. In particular $\pi_2(G/T)$ is free, and by the LES of a fibration $\pi_2(G)$ injects into $\pi_2(G/T)$ and is torsion free, so zero.