For many years there was never any rough at our course. The fairways(loosely used term) were mown at 3/4's of an inch, wall to wall. Then we convinced our super to lower the cut, which he did. We then found that a 2" cut of rough actually benefitted players by keeping their errant shots from rolling into greater trouble, as our terrain is severe in certain places.
This is a case where deeper grass enhanced the playability of our course.