Let $X$ be a complex variety of dimension $m$ and $D_1,D_2,\dots, D_m \subset X$ irreducible divisors with $D_i \neq D_j$. Can we have $D_1 \cdot D_2 \dots \cdot D_m < 0 $ ?
The only concrete example of negative self-intersection I know is when $m=2$, and when $D_1 = D_2$ is a curve in a surface, for example the exceptional divisor of a blow-up.