Big Timber is a small south-central Montana town where the Yellowstone River valley opens toward the Crazy Mountains and the Boulder River country. It is not a polished resort town, and that is part of the appeal: you get a real ranch-and-river community with enough nearby landscape to fill a spare day well.
The town itself has a compact historic centre and the useful Crazy Mountain Museum. The better reason to stop is the wider Sweet Grass County setting: Natural Bridge Falls south on the Boulder River, Greycliff Prairie Dog Town State Park east along the old highway, and trout water in almost every direction.
This is a good one-day detour between Livingston and Billings, or a quieter base if Bozeman's growth has worn you out. Come for open country, short sights, and that particular Montana feeling of town streets ending quickly in mountains and grass.