Assume that a language $L$ is given. The alphabet is $\{1,2,3\}$ and the machine accepts all of the strings, in which the last symbol is repeated at least two times and there is no "bigger" symbol between these repetitions.
For example, the strings $3212113, 123113, 2112, 11$ is accepted, but $123, 232, 12312$ is not!
Question: Design a NFA for this language!
Note: I've seen many other examples. In those ones, the language was explicitly given by a regular expression or something like that... But in this case, I cannot write the regular expression of the strings which are accepted. So, I can't design the NFA, too!
Hint. Let $A = \{1,2,3\}$. Your language is the union of the three following languages \begin{align} L_1 &= A^*3A^*3\\ L_2 &= A^*2\{1,2\}^*2\\ L_3 &= A^*11 \end{align} Now, you can compute a NFA or even a DFA for each of these languages (actually the minimal DFA of each of them has exactly 3 states and should be easy to compute). Then you should be able to find a NFA for $L_1 \cup L_2 \cup L_3$.