My question is: given an abelian group $G$ with addition $+$, is there some natural multiplicative structure that arises so that we can define a ring $(G, +, \cdot)$. For instance, multiplication on $\mathbb{Z}$ and $\mathbb{Z}_n$ are entirely determined by addition, since it must be that $ma = a + \ldots + a$, where the addition is $m$ times.
For finitely generated abelian groups $G$, we know that its representation according to the Fundamental Theorem of Finitely Generated Abelian Groups is
$$G = \mathbb{Z}_{p_1^{r_1}} \times \ldots \times \mathbb{Z}_{p_n^{r_n}} \times \mathbb{Z} \times \ldots \times \mathbb{Z}$$
Since each of those factors has a natural ring structure, we can define a ring structure on $G$ as the product of these ring structures. That leaves the question: can we define a "natural" ring structure on infinitely generated abelian groups $G$?
Expanding on egreg's comment.
The Prüfer group for $p$ has the property that all elements are of finite order, and there is an element of order $p^k$ for every $k$.
Now, in a ring, every element must be of additive order equal to a divisor of the additive order of $1$, because if $n1=0$ then $nr=n(1r)=(n1)r=0$ for any $r\in R$. So the Prüfer group cannot be made a ring.
On a side note, your insistence on using the word "natural" muddles the question considerably, since the word "natural" has a very specific meaning in category theory, and none of the constructions you've defined above are "natural" in the sense of category theory.