Directed graph where DFS returns on a node before all its child nodes are visited? Give an example of a directed graph in which a depth-ﬁrst search backs
  up from a vertex $v$ before all the vertices that can be reached from
  $v$ via one or more edges are discovered.


My professor recently asked this question as a warm up to lecture, but never answered it. I still have not figure how that is possible. Why would it return if it's not complete?

I just can't see a scenario where this would happen.

It would never return, since DFS is (essentially) recursive and it can't return without having hit all base cases.
Consider the following graph. If DFS will first go to the second node, it will back up even though the third node is reachable from the second.