How weird can a ring spectrum be if all of its modules are free? Let $R$ be a ring spectrum. If $\pi_\ast(R)$ is a graded field, then all module spectra over $R$ are free. But I don't believe the converse holds. How badly can it fail?
I'm assuming that $R$ is at least an $A_3$ ring spectrum, and probably $A_\infty$ (I'm also interested in more highly-structured cases); if the answer depends on just how highly structured $R$ is, I'd find that interesting.
Note that if $\alpha \in \pi_\ast(R)$ is non-nilpotent, then the freeness of the localization $\alpha^{-1} R$ entails that $\alpha$ is a unit. So if all module spectra over $R$ are free, then $\pi_\ast(R)$ is a local ring of dimension 1. But $\pi_\ast(R)$ might have nonzero nilpotents as far as I can see.
Question: What is an example of a ring spectrum $R$ such that $\pi_\ast(R)$ has nonzero nilpotents and yet every module spectrum over $R$ is free?
 A: I think what you are looking for is the notion of a semisimple ring spectrum, as studied by Hovey and Lockridge. For such a ring spectrum, $E$, every module spectrum is projective, i.e. is a retract of a coproduct of suspensions of $E$. The homotopy groups $E_*$ of such spectra are characterized in Theorem 1.2 of this paper: $E_* \cong R_1 \times \dots \times R_n$ where $R_i$ is either a graded field $k$ or an exterior algebra $k[x]/(x^2)$ over a graded field with a unit in degree $3|x|+1$. Such rings $E_*$ are called graded commutative $\Delta^1$-rings, and clearly have plenty of non-zero nilpotent elements.
If $E$ is commutative, then $E$ is semisimple if and only if $E_*$ is a graded commutative $\Delta^1$-ring and for every factor ring of the form $k[x]/(x^2)$, we have $x\cdot \pi_*(C) \neq 0$ where $C$ is the cofiber of $x\cdot E$. 
I'll finish by mentioning classic work of Hopkins and Smith, that the OP is surely aware of but future readers might not be, that defines the fields of stable homotopy theory as ring spectra $E$ such that $E_*X$ is a free $E_*$-module for all spectra $X$, and that characterizes fields as spectra having the homotopy type of a wedge of suspensions of Morava $K$-theory $K(n)$ for some fixed $n$ and prime $p$.
