Antarctica is a great idea as well. I am not sure what the political challenges would be if somebody with a sufficiently absurd amount of money to throw at the project actually attempted it. The antarctic treaty is supposed to prohibit nation states from claiming pieces of it as sovereign territory, but those negotiations from the 1950s/1960s didn't really envision modern billionaires doing such things independently.

Whether somebody would have the early stages of their construction project shut down by a flown-in team of military forces from some major nation states (US, Russia, UK) is another question. Or simply blockaded and shut down at places like the ports in Chile, Argentina, Tasmania, mainland Austrlia, New Zealand, South Africa.