Let $\mathscr{C}$ be a Waldhausen category, i.e. a category pointed by $0$, with cofibrations and weak equivalences. Recall a map $B \to B/A$ in $\mathscr{C}$ is called a quotient map if it is the pushout of $A \to 0$ along a cofibration $A \to B$. Apparently, the class of quotient maps in $\mathscr{C}$ is not necessarily closed under composition, as stated in the original paper by Waldhausen and reiterated in later literature. Unfortunately, I have not found an example of a Waldhausen category for which the class of quotient maps is not closed under composition. For the life of me, I cannot seem to construct one myself.
I think anything `pointed' has little chance of working: pointed sets, pointed spaces, relative CW-complexes... To see this, let $A \to B \to B/A$ and $C \to B/A \to (B/A)/C$ be cofibration sequences in some Waldhausen category of pointed objects. Then let $D$ be the inverse image under the map $B \to B/A \to (B/A)/C$ of the point $*$. Now if $D \to B$ is a cofibration, then $D \to B \to (B/A)/C$ is a cofibration sequences, hence the composition $B \to (B/A)/C$ is again a quotient map...