Cyclic deterministic finite automata, exponential advantage over DFA
Let $\Sigma$ be an alphabet of size $n$, and consider the language $L_n$ of all words over $\Sigma$ which contain all letters of the alphabet. It is well-known that the minimal DFA for $L_n$ contains $2^n$ states. On the other hand, $L_n$ is accepted by a CDFA with $n+1$ states.