Questions like this are well studied by homotopy theorists, in particular those working in Lusternik-Schnirelmann category (**warning:** nothing to do with the objects and morphisms kind of category!). In particular, what you are asking for is examples of spaces with weak category equal to 2. The *weak category* of a pointed space $X$ is the least $k$ such that the reduced diagonal $\triangle_k\colon X\to X\wedge X\wedge\cdots\wedge X = X^{\wedge k}$ is null-homotopic. The weak category was introduced by Berstein and Hilton in this paper, as an approximating invariant for LS-category: note that $\mathrm{wcat}(X)\leq\mathrm{cat}(X)$ for any space. (**Warning 2:** Nowadays, the prevailing convention seems to be that these invariants are normalised, so that the weak category of a contractible space is zero.)

Note that the weak category of any (non-contractible) suspension $\Sigma Y$ is $2$. In the paper of Berstein and Hilton linked above they construct spaces $Z$ with weak category $2$ and LS-category $3$ (which therefore cannot be suspensions).