Nick Hexum's name will probably only be familiar to you if you're a fan of heavy rock and alternative music; he's been the singer and guitarist for the rock band 311 since 1988. Even though his music has more appeal to niche interest groups than a larger audience, he still made enough money from it to purchase Melody Key; one of the Florida Keys. It's only 5.5 acres of land in total, but as of 2012, it was ranked as the world's fifth most expensive island.
The island is completely off the grid, and has changed hands several times in the past few years between various different millionaires, all of whom have performed upgrade and maintenance work on it. It most recently went on the market in 2017, with an asking price of $7m. You don't get a beach with the island, but you can at least enjoy some close-up views of iguanas.