There is a Proof due to Stasheff in "H-space from homotopy point of view" (Theorem 6.7). The argument is fairly simple to describe. $\def\OP{{\mathbb O\mathbf P}}$
If $S^7$ admits a homotopy associative multiplication, then one should be able to construct $\mathbb{O}P^3$$\OP^3$. It would follow that $\tilde{H}^*(\mathbb{O}P^3;\mathbb{Z}/3)$$\tilde{H}^*(\OP^3;\mathbb{Z}/3)$ is generated by $u_8$ in degree $8$ with the relation that $u_8^4 =0$. It follows that, $$ P^4(u_8) = u_8^3. $$ However, $P^4 = P^1P^3$, which means the $u_8^3 = P^1x$, where $x \in \tilde{H}^{20}(\mathbb{O}P^3;\mathbb{Z}/3) = 0$$x \in \tilde{H}^{20}(\OP^3;\mathbb{Z}/3) = 0$. Thus $u_8^3 = 0$ which contradicts the existence of $\mathbb{O}P^3$$\OP^3$ and consequently the existence of homotopy associative multiplication on $S^7$.
More interestingly this proof suggests that the obstruction to homotopy associative multiplication is $3$-local. This leads to the following question.
Question: Is $S^7_{(2)}$ homotopy associative? Does the reference to James work in the answer due to Jon Beardsley, addresses this question? (Tyrone's comment made me think of this question.)
(Sorry for asking a question inside an "answer".)
