What's an (explicit) example of nontrivial almost ring $(R,+,*)$ in the sense that $+, *$ follow the usual distributive laws,it forms semigroup (i.e. we don't assume R has identity, see note) under multiplication and non-abelian group under addition?
By nontrivial I mean that the multiplication is not trivial, that means, it is not the case that $ab=0$ for all $a,b\in R$.
Note: If $R$ has an identity, then $(R,+)$ is necessarily commutative, because expanding $(1+1)(a+b)$ in two ways using distribution yields $a+a+b+b=a+b+a+b$, thus $a+b=b+a$.
The same argument without identities still shows that
$$(a + b)(c + d) = ac + ad + bc + bd = ac + bc + ad + bd$$
hence that $ad + bc = bc + ad$ for any $a, b, c, d \in R$. In other words, commutativity holds for any elements in the image of the multiplication map $R \times R \to R$ (which in the presence of identities is always all of $R$). So the only way to break commutativity is to make the image of the multiplication map small somehow.
An easy way to do this is to take the direct product $R = G \times S$ of a nonabelian group $G$ equipped with the zero multiplication and a nonzero ring $S$ with identity.