minimum number of states in cross product of two minimum DFAs
You probably know that if $\cal A$ and $\cal B$ are DFA, then their product recognizes the language $L({\cal A}) \cap L({\cal B})$. Now if ${\cal A} = {\cal B}$, then $L({\cal A}) \cap L({\cal B}) = L({\cal A})$. Thus if $\cal A$ is a minimal DFA with at least two states, ${\cal A} \times {\cal A}$ is never minimal.