I assume you meant there to be a “not” in the first sentence?
Yes. It’s NOT against the law in the United States to pretend your living room is an independent country, nor is it against the law for you to declare your couch fort an independent country while your sister declares the easy chair fort a separate independent country. Nor is wearing a crown, or telling everyone to call you Princess Sparklepony the First.
Which reminds me: did you ever get any money from the county for that wrongful arrest?
Me?
There’s lots of unclaimed territory in Antarctica just waitin’ to be settled. As long as you like the taste of penguinburgers.
What? Eat my subjects?
It’s “claimed” by international treaty as being more or less unclaimable. Australia or somebody would boot you out. If they didn’t, the US & Russia probably wouldn’t mind doing it.
No. Let’s try that again:
Picture the Chief of Police writing the above words on the blackboard 100 times.
You’d only need 1 Billionaire to stage a hostile takeover of Pitcairn Island.
Oops.
Antarctica is claimed as in actually claimed, and the same bits are claimed by more than one country. It is definitely an issue, just not an issue that people have wanted to spend much money on.
AIR The international treaty is that the various claimants aren’t going to station military, build borders, or do anything destructive like atomic bomb tests down there.
There is actually a substantial piece of Antarctica that hasn’t been claimed by anyone. But a.) no one will recognize anybody claiming that area (or any other Antarctic territory) and b.) it’s in bloody Antarctica.
That’s okay. Apparently it was a bad joke.
Freedom loving businessmen try this from time to time. Robert Vesco, accused of securities fraud and later drug trafficing by the US government, attempted to buy the island of Barbuda from Antigua during the 1970s. The Caruana-Cuntrera clan, hailing from Sicily, allegedly purchased 60% of the island of Aruba, though unlike Vesco they did not try to arrange sovereignty: they would have been satisfied with political control.
Vesco died outside of prison. Pasquale Cuntrera and Alfonso Caruana were not as fortunate.
So to answer the OP’s question, if you have a cool $200 million on tap have your people talk to my people. 
In 1846, James Brooke persuaded the Sultan of Brunei to make him Rajah of Sarawak. His family maintained their status for over a century. The Clunies-Ross family acquired title to the Cocos Islands in 1886.
That’s not likely to happen today. Any democratic government would lose the next election if they parted with any territory. Your best bet would be to find some dictator who could be bribed to go along with your scheme. Preventing him from welching on the deal after he cashed the check would be problematic, though.
Only fifty countries have signed the Antarctica treaty, though. Play your cards right and you might get recognition from the rest.
Well, Brunei is still not democratic, but now has jillions of dollars and is unlikely to give up additional territory. You could probably get the governments of Oman or Yemen to cede bits of the Rub’ al Khali if you offered enough money.
Would countries with large amounts of poor be vulnerable to a peaceful takeover? An example might be Kenya. Several billionairs could go in along with some charismatic speakers such as Obama for instance and through a combination of bribery and campaining win the support of the people and the existing military. With grandiose plans for industry, healthcare, housing, agriculture and infrastructure I can see this as not being so far fetched.
I don’t see it being much different than the communist push we saw in the 50’s and 60’s throughout south and central America as I doubt everyone would join peacefully but I think it is a viable possibility.
This is speculative, but our ambitious billionaires could model themselves on Hong Kong (one country, two systems) or Singapore (a city state which peaceably separated from Malaysia).
My favored candidate for a [del]failed state[/del] utopian paradise is the Western Sahara. It’s basically unviable on its own, but they don’t want to be ruled by Morocco. Surely the [del]terrorists, thugs and colonialists[/del] resident politicians would be willing to part with a sliver of it, if a strong case could be made [del]with bribery[/del] for freedom.