I have been told there cannot be a consistent theory defining a distribution product. Googling for information, I found 1 and 2. Number 1 gives interesting hints on what might happen, and defines a convolution for tempered distributions. Number 2 precisely states the "Schwartz impossibility result", which says:
There is no associative algebra over $\mathbb{R}$ containing $\mathcal{D}'$ as a vector subspace and the constant function 1 as unity element, having a differential operator acting like the differential operator on $\mathcal{D}'$ and the algebra multiplication of continuous functions is like their pointwise multiplication.
I was unable to find a proof of this result, but that is another question. This question wants to focus on: what if I take another distribution as the unity? The $\delta$, for example? THe example at number 2 shows that even then we couldn't keep a natural product between functions and distributions, but would it be possible to define an associative distribution product with anything as a unit? If not, why?
PS OK, it might not be interesting to define one, but just for curiosity's sake, could it ever be possible?
Update: As Giuseppe Negro notes in a comment, if we restrict ourselves to distributions whose supports are "bounded from the left", for example contained in $[0,+\infty)$, then the convolution is associative and commutative. This is a result proved in the book by Friedlander and Joshi, on p. 57, in §5.3. But what happens without that bound? What goes wrong with distributions with two-side unbounded support?
