Some nations are more well known than others, especially when it comes to lost countries.
In the golden age of piracy a group of the most notorious privateers decided to found their own country ion a small island. It came to be known as the pirate republic, with Nassau it's capital. But being populated by pirates, the British empire soon absorbed it. Like most other countries on this list, the pirate republic was short lived.
The ezo republic was another island nation. This time founded by forces on the losing side of a Japanese civil war. With the emperor of Japan consolidating power, they fled to Ezo and attempted to found a new democratic state.
The golden horde was not short lived. It broke away from the mongol empire and long subjugated many different peoples. The golden horde survived so long that it's ruling class changed in ethnicity. Another Asian kingdom was established during the Taiping rebellion. It was a christian nation where all people were equal under national law. Their leader claimed to be the brother of Jesus Christ, and therefore divine right to rule.