Maybe this belongs in pre-1900, but I think it's definitely doable in post-1900. The challenge is to create a country, that is to say a geopolitical entity that is recognized as a sovereign state by most of the rest of the world, and maybe has a seat at the UN/regional organizations, that is effectively the property of one person or a company. An example that sort of comes to mind is the White Rajahs, but I'd like whatever country is created to persist to the present day.
I was thinking perhaps some religious group could pick an island in the Caribbean or pacific and basically subsume the population and government, like what they tried to do on the county level with Rajneeshpuram. A company that effectively buys out the entire population of a country and takes over the state for its own purposes will also work. IOTL there's definitely companies that could do that, but to my knowledge none have ever done so.