“There are few places in Yellowstone NP to ride a mountain bike--the Mount Washburn/Chittenden Rd. route is one.”
— Chris Stewart
The road is only open when this part of the park is open, and even then, I would imagine elevation snow dictates how accessible the summit is.
The scenery here is incredible; the workout is great; you may even bump into a mountain sheep, buffalo or even bear while riding up or down. Can't beat it!
After a couple of miles up Chittenden Rd., there is a bathroom, parking, and the road is gated. It starts out at about 8,500 feet.
Ride around the closed/locked gate and begin the three-mile climb to 10,243 feet--over 1,700 feet of elevation gain. It is not particularly steep; however, when you take into account the altitude, it's a workout. Nothing technical about it. There are some larger, loose rocks in places, but they are easy enough to avoid.
After reaching the top, take a scenic pic or selfie--now for the fun part. There is a lookout tower at the top with a restroom and viewing room. Plenty of people up there to take your pic next to the Mount Washburn sign. The only thing to remotely worry about is the amount of hikers on the road. Just be respectful.
With the workout, the scenery and the wildlife, this ride is more than worth it.