Is it possible to define a DFA for the language of words having exactly one double letter? Consider the language $\{ w \in \Sigma^*~|~ w~\text{contains exactly one double letter}~\}$. For example,
$baaba$ has exactly one double letter, but $baaaba$ has two double letters.

Suggested answer for the case where $\Sigma = \{a,b\}$ (even if it's an RE not a DFA): $(\epsilon \cup b)(ab)^*aa(ba)^*(\epsilon \cup b) \cup (\epsilon \cup a)(ba)^*bb(ab)^*(\epsilon \cup a)$.
In general, if you claim to have found a regular expression for the language, you can follow Thompson's algorithm to convert that expression into an NFA and then using subset construction to convert that NFA into a DFA.