Not sure where to go with this... I've done the following, though:
$A$ and $C$ must be square matrices if they are invertible. Let $A$ be an $m\times m$ matrix, and let $C$ be an $n\times n$ matrix. Let $B$ be a $m\times n$ matrix so that $ABC$ is well defined.
$A$ has a rank of $m$ and $C$ has a rank of $n$. $rank(AB)\leq m$ and $rank(AB)\leq rank(B)$ Also, $rank(BC)\leq n$ and $rank(BC)\leq rank(B)$.
From here, I have no idea what to do. I don't even know if I'm on the right track. The best I can figure is that $rank(ABC)\leq \min(rank(A),rank(B),rank(C))$, but I have no idea how to show that $rank(ABC)=rank(B)$.