I wonder whether are there known algebraic systems where there are non-infinity (not satisfying $x+a=x$) elements such that their power or product is infinity (an algebraic object that satisfies $x+a=x$, like in extended reals)?
I can think about $\overline{\mathbb R}^2$, which seemingly has divisors of infinity like $(1,\infty)$, but what about those with elements that squared give infinity?
Such structure $(S,+,\cdot)$ exists.
Take any non trivial ring $A$ with unit having an element $e\in A$ such that $e^2=0_A$.
Set $S=A$, $+=\cdot_A$, $\cdot=\cdot_A$, $\infty=0_A$.
Then for all $a\in A$, $a+\infty=a\cdot 0_A=0_A=\infty$, and $e\cdot e=e^2=0_A=\infty$. However, not all element is $\infty$. Indeed, $1_A\neq \infty$. Otherwise, $1_A=0_A$ and $A$ would be trivial.
As a concrete example, you may take $A=\mathbb{Z}/n^2\mathbb{Z}$, and $e=\bar{n}$ (there are other examples as well in the same spirit)
However, I don't understand why you are looking for such structures. They have extremely limited interest from a computational point of view. For example $(S,+)$ cannot be a group (because of the existence of $\infty$), and I am pretty sure that you cannot have distributivity properties, even if a proof does not come into my mind right away...