While much of the Grand Teton National Park is paved, there is a rare off-roading opportunity on the River Road. Join buffalo and elk along the Snake River. This 12-mile gravel and boulder route winds through tall grasses and sagebrush with clear views to spot wildlife and the Grand Teton Mountain Range constantly visible in the background.
Though we did spot a compact car attempting the route, I have a feeling they turned around at a certain, steep boulder covered descent. Otherwise, any rig with moderate clearance and 4-wheel drive should be fine, weather permitting.
UPDATE: As of June, 2017, the River Road has been closed as portions have crumbled into the Snake River.