Another perspective of the question is from the point of view of computational complexity. According to the this paper (https://arxiv.org/pdf/1304.7705.pdf), for a simply connected finite simplicial complex $X$, computing $\pi_n (X)$ is a computationally very hard problem with respect to the parameter $n$ ($W[1]$-hard to be concrete).

It has been known for a long time that higher homotopy groups are algorithmically computable (the first results go back to Brown), and later it was shown (https://arxiv.org/pdf/1211.3093.pdf) that there is an algorithm which for a finite simply connected simplicial complex $X$ and $n \geq 2$ computes $\pi_n (X)$ in time, polynomial in the size of $X$. However, in this polynomial result $n$ is taken as part of the input. The result I stated above is that if you vary $n$, then the complexity of the problem explodes. I find this point of view particularly nice because it gives some concrete feeling for how complicated the computation of higher homotopy groups is (including the homotopy groups of spheres). On the other hand, one may point out that many results have been obtained regardless of the complexity but the sophistication of the tools required to obtain those results is yet another instance of this complexity.

Hope that is useful.