I'll be working in the stable world. It's an easy observation that any 2-cell complex (over the sphere) with bottom cell in dimension zero is a Thom spectrum: any such complex is the cofiber of some element $\alpha\in \pi_n S$, so it is the Thom spectrum of the map $S^{n+1} \to B\mathrm{GL}_1 S$ classifying $\alpha$. Is there an example of a finite complex (again, with bottom cell in dimension zero, which is a necessary condition) which is *not* a Thom spectrum? Is there such a 3-cell complex?