When Time magazine last week named Elon Musk “Person of the Year,” it painted a spartan picture of his living arrangements. “The richest man in the world does not own a house,” the piece began.

The magazine was repeating what has become a core element of Mr. Musk’s public persona. Since last year, when the billionaire Tesla Inc. and SpaceX CEO announced he was “selling almost all physical possessions” in an effort to simplify his life, Mr. Musk has repeatedly portrayed himself as a man of modest tastes.

“My primary home is literally a ~50k house in Boca Chica/Starbase that I rent from SpaceX,” he tweeted this June, referring to a Texas town near the Mexican border near SpaceX’s rocket-launch facility. “It’s kinda awesome though.”

What Mr. Musk hasn’t said, and what few people know, is that for roughly a year he has also been living in a waterfront estate in Austin owned by a rich friend nicknamed “Kenny,” people familiar with the matter say—a home so extravagant that it was the most expensive listed in the Texas capital when it was sold just a few years ago.