I only know through stories that mod 3 moore spectrum is not associative. I do not know of any proof. I have been informed that Toda had proved it in the paper "Extended $p^{th}$ power". I was not able to follow it. Can anybody give me a proof that mod 3 Moore spectrum is not associative?

On spectra realizing exterior parts of the Steenrod algebra(sciencedirect.com/science/article/pii/0040938371900176), which you might find helpful. (I don't know if this proof is any different from the one in the Toda reference you mention, which I don't have available.) – Eric Peterson Jan 15 '13 at 20:47