For two based topological spaces $(X,x_0)$ and $(Y,y_0)$, the smash product $X\wedge Y$ is the quotient $(X\times Y)/(X\vee Y)$, where $X\vee Y$ is the subspace $(X\times\{y_0\})\cup(\{x_0\}\times Y)$.
On the category of based topological spaces, the smash product does not associate. Counterexamples are not super easy to come by and prove, but the most well-known can be found here.
Now among spaces which are either locally compact and Hausdorff, or which are compactly generated, the smash product is associative. In particular, this covers all CW complexes and manifolds, which is nice. In addition, via CW approximations, we can at least see that
$$\pi_n((X\wedge Y)\wedge Z)\cong \pi_n(X\wedge(Y\wedge Z))$$
for all $n$. However, I don't see how to upgrade this to even a weak homotopy equivalence in general. My question is the following.
For any pointed topological spaces $X,Y,Z$, is $X\wedge(Y\wedge Z)$ homotopy equivalent to $(X\wedge Y)\wedge Z$?