In motivating $A_\infty$-spaces to my students I'm going to insist on the homotopy invariance of the notion, saying that "being $A_\infty$ is the homotopy invariant version of being a topological monoid" and to stress this I'd like to say that if $X$ is a topological monoid and $Y$ is a space homotopy equivalent to $X$ then $Y$ will carry an $A_\infty$-structure making it equivalent to $X$ as an $A_\infty$-space, but in general not a topological monoid structure with this property. But at this point I see to my shame that I miss an explicit example of this!
Clearly the most dramatic example would be that of a space $Y$ which is homotopy equivalent to a topological monoid $X$, but such $Y$ carries no topological monoid structure at all, not to have to go into the equivalence issue. For a while I thought the closed interval could be an example of this (double shame: there are at least two very simple and well known topological monoid structures on $[0,1]$!), so I'm completely without examples, and I do not either know if such a space $Y$ does actually exist at all.
edit: despite I originally formulated my question in the most dramatic possible form, an example where, given a homotopy equivalence $f:Y\to X$ there is no monoid structure on $Y$ such that $\pi_0(f)$ is an isomorphism of monoids $\pi_0(Y)\to \pi_0(X)$ is even better for what I need to explain, namely that going from monoids to $A_\infty$-spaces not only $Y$ is naturally endowed with an $A_\infty$-space structure, but $f$ is promoted to an equivalence of $A_\infty$-spaces. So I will now leave the original question open as a general topology question which may have its interest in its own (despite it is admittedly an odd question), while for myself I'll be perfectly satisfied with the very nice answer by Tyler below.