Let's follow your intuition:
There's intuition, the intersection is always the "smallest of the
sets" so eventually it will be $(\infty,\infty)$ or something like
Okay, but $(\infty,\infty)$ doesn't have any elements in it. Assuming we're dealing with the domain of real numbers, $\infty$ isn't one. (Recall that in real numbers the symbol $\infty$ isn't a number; it's always a shorthand placeholder for some statement about unbounded limits.)
On the other hand, even if we were talking about the extended real numbers where $\infty$ is an element, the notation $(\infty,\infty)$ indicates that endpoints are not included; that being $\infty$, so again this is an "interval" with no points in it.