None, really. The nearest town of any size is Roanoke, Virginia.
The nearest bona fide city to the Greenbrier is Charlotte, North Carolina. However, Charlotte is over 200 miles south of the resort.
Make no mistake - the Greenbrier is an old-school resort that is far from most large towns. It has existed as a getaway for its entire life, and one must "get away" to go there.