For any integer $d\ge 1$, I would like to find a function $f\in L^1(\mathbb{R}^d)$ such that for any ball $B\in \mathbb{R}^d$, $f$ is not essentially bounded on $B$.
If you solely asks that $f$ be essentially unbounded on any ball containing a point, the origin, for example, then the function $|x|^{-d+\varepsilon}\chi_{|x|\le 1}$ for $0<\varepsilon<1$ is integrable but unbounded on any such ball. However, I'm not sure how to come up with a function that is essentially unbounded on any ball.