You're going to get many different answers depending on the tastes of the topologist answering...

I like to think about homotopy groups of spheres through framed cobordism. Theories like unoriented and complex cobordism are understandable for a couple reasons. Technically they are calculable because we can understand their cohomology so well over the Steenrod algebra. But morally they are understandable because they are amenable to analysis through characteristic classes. But for framed bordism, the structure group is the trivial group. So either the theory is going to be trivial, or really hard because there are no characteristic classes to use. It turns out that it is the latter.