I wondered this, and tried to find an answer online, but the only thing I could find was a statement that the set of functions which are in all $L^p(\mathbb{R}^n)$ is well-studied. But what functions are in all $L^p(\mathbb{R}^n)$ spaces?
If the answers are very different, I’d be interested in both the $p<\infty$ and the $p \leq \infty$ case.
As an application of interpolation (see here for a related theorem, or use Holder's inequality), this is just the set $L^1(\mathbb{R^n}) \cap L^{\infty}(\mathbb{R^n})$. So any bounded, integrable function is in every $L^p$ and vice-versa.
So the moral of the story is that the endpoints tell you everything.