Let $\pi: Y \to X$ be a proper birational morphism.
Let $E$ be the exceptional locus, and $Z = \pi(E)$ the scheme theoretic image (so that it is closed). I want $X,Y,Z,E$ to be smooth.
Now take $\pi^{-1}(Z)$, this is supported on $E$ and so can be written $\sum a_i \cdot E_i$.
What is an example where $a_i > 1$?
Note this isn't the same as the pullback of the canonical bundle having nonreduced support (which we can make via blowing up the plane twice)
Bonus points for toric answers which teach us to be comfortable with toric geometry!
If $E$ is not smooth but a union of smooth thing it must not be reduced.
Blowup $P^2$ at $p$ to get $E_1$ . Then blowup at $q \in E_1$, then at $E_1 \cap E_2$ to get $E_3$ which will have multiplicity $2$.