The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero.

Question: What are some "toy models" for the ring $\pi_\ast^s$?

By this, I mean examples which illustrate particular pathologies of this ring. For example, if $k$ is a field, then the ring of dual numbers $k[x]/x^2$ exhibits the pathology of being non-reduced. But that barely scratches the surface of the bad properties that $\pi_\ast^s$ has.