He’s on the patio. He’s on the golf course. Everywhere Donald Trump looks, there is the world’s richest man.

In nearly every meeting that President-elect Donald J. Trump holds at Mar-a-Lago, alongside him is someone who has been elected to nothing, nominated to nothing and, only a few months ago, had no meaningful relationship with him.

Elon Musk.

The world’s richest person has ascended to a position of extraordinary, unofficial influence in Mr. Trump’s transition process, playing a role that makes him indisputably America’s most powerful private citizen. He has sat in on nearly every job interview with the Trump team and bonded with the Trump family, and he is trying to install his Silicon Valley friends in plum positions in the next administration.

Mr. Trump announced on Tuesday that Mr. Musk would help lead what he called the Department of Government Efficiency, a new body to “dismantle government bureaucracy.” But Mr. Musk’s true influence on the Trump transition effort goes well beyond that posting.

Mr. Musk has assumed an almost mythical aura in Mr. Trump’s inner circle. At Mar-a-Lago one recent evening, he walked into the dining room about 30 minutes after the president-elect did and received a similar standing ovation, according to two people who saw him enter.

Mr. Musk, often with his 4-year-old son X on his lap, has spent most of the last week at Mar-a-Lago, joining not just interviews but almost every meeting and many meals that Mr. Trump has had. He briefly shuttled back to Austin, Texas, where he has a $35 million compound, before returning on Friday, where he ate in Mar-a-Lago’s dining room and on its patio, roamed the gift shop and spent time on the golf course — all alongside the president-elect.

“I’m happy to be the first buddy!” he replied to a social-media follower this weekend.

This article is based on roughly a dozen interviews with Republican donors, politicians and friends of Mr. Musk, many of whom insisted on anonymity to talk about private conversations.

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