I used to think that the entire theory was intellectual masturbation, but two examples in particular completely changed my mind.
The first is the Pontryagin-Thom construction, which exhibits an isomorphism between the $k$th stable homotopy group $\pi_{n+k}(S^n)$ and the framed cobordism group of smooth $k$-manifolds. This is even interesting (though more elementary) in the case $k = 0$, where it recovers the basic degree theory that you learn in your first course on topology. This was originally developed by Pontryagin to compute homotopy groups of spheres, but now it is regarded as a tool in manifold theory. These matters are discussed in Chapter 3 of Luck's book on Surgery theory, for example.
The second application is to physics. Unfortunately I don't understand this story very well at all, so I'll begin with what I more or less DO understand (which may or may not be well-known). The basic idea begins with the problem of situating electromagnetism in a quantum mechanical framework. Dirac began this process by imagining a "magnetic monopole", i.e. a particle that would play the role for magnetic fields that the electron plays for electric fields. The physical laws for a charged particle sitting in the field determined by a magnetic monopole turn out to depend on a choice of vector potential for the field (the choice is necessarily local), and Dirac found that changing the vector potential corresponds to multiplying the wave function $\psi$ for the particle by a complex number of modulus 1 (i.e. an element of U(1)). If we think of the magnetic monopole as sitting at the origin, then these phases can naturally be regarded as elements of a principal $U(1)$-bundle over $M = \mathbb{R}^3 - \{0\}$. But $M$ is homotopy equivalent to $S^2$, and principal $U(1)$-bundles over $S^2$ are classified by $\pi_1(U(1)) = \mathbb{Z}$. Proof: think about the Hopf fibration. The appearance of the integers here corresponds exactly to the observation of Dirac (the Dirac quantization condition) that the existence of a magnetic monopole implies the quantization of electric charge. It is remarkable to note that Hopf's paper on the Hopf fibration and Dirac's paper on magnetic monopoles were published in the same year, though neither had any clue that the two ideas were related!
The story goes on. The so-called "Yang-Mills Instantons" correspond in a similar way to principal $SU(2)$ bundles over $S^4$, which are classified by $\pi_3(SU(2)) = \mathbb{Z}$. Again, the integers have important physical significance. So these two classical examples motivate the computation of $\pi_1(S^1)$ and $\pi_3(S^3)$, but as is always the case this is just the tip of an iceberg. I am not familiar with anything deeper than the tip, but I have it on good authority that physicists have become interested in homotopy groups of other spheres as well, presumably to classify other principal bundles (it seems like a bit of a coincidence that the groups which came up in these examples are spheres, but maybe one reduces homotopy theory for other spaces to homotopy theory for spheres). People who know more about physics and/or the classification of principal bundles should feel free to chime in.
A great reference for the mathematician who wants to learn something about the physics that I discussed here is the book "Topology, Geometry, and Gauge Fields: Foundations" by Naber.
You said you don't want to talk about framed manifolds, but that's a good way of seeing the 24. $\nu$ is represented by $SU(2)$ in its invariant framing. Take a K3 surface. It's framed, and it has Euler characteristic 24. Take a vector field that has 24 isolated zeroes of index 1. If you cut out a little disk around each of these 24 zeroes, the boundary will be an $S^3 = SU(2)$ with its invariant framing. So the K3 surface minus these 24 little disks is a null-bordism of $24\nu$. Probably not suitable for your course, as you would have to explain framed bordism and K3 surfaces, but cute nonetheless I think. By the way, the analog for $\eta$ is the two-sphere (Euler characteristic 2).
Best Answer
Computing $\pi_\ast(S)$ is a tedious business that to this day can only be done "by hand", i.e. by humans. The $p=2$ computation up to dimension 64 was completed by Kochman (see his SLNM book) with later corrections by Kochman/Mahowald. This was mainly (but not exclusively) based on the Atiyah-Hirzebruch spectral sequence $$H_\ast(BP,\pi_\ast(S)) => \pi_\ast(BP).$$
The available approximations to $\pi_\ast(S)$ try to decompose the problem into two steps:
computation of the approximation, e.g. the $E_2$ term of a spectral sequence.
computation of the differentials.
It's probably not suprising that step 2 requires human intervention; but often even the first step is a difficult computational challenge: for example, nobody seems to know how to compute the $E_2$-term of the Novikov spectral sequence efficiently.
Since this $E_2$-term is the cohomology of the moduli stack of one-dimensional formal groups, this problem should appeal to number theorists as well. And although number theory has a strong computational branch it seems that not much has been done here.