>>1745252Mongolia. Honestly it's one of my favorite trips. There are hardly any fences (Mongolia is like 90% public land), the people are super friendly, and Mongolian horses are a blast to ride. If you're not an advanced rider I wouldn't suggest going out by yourself as the horses are extremely independent and finnicky, so you should be well able to control them before setting off on your own. Get a guide to help you out for a few days (you'll need one anyway to get a horse, saddled, etc.) I suggest a company called stepperiders.
http://stepperiders.mn/ 