The creators of Elon GOAT say the name of their cryptocurrency was inspired by their respect for Mr. Musk. They and his other fans think he is the “greatest of all time,” or a “GOAT.” They took the admiration literally, spending $600,000 to create a sculpture of Mr. Musk’s head, wearing a gold-plated dogecoin necklace on a goat’s body. The rocket can move, pointing to the sky as if it is taking off. Gas lines run through it so that flames can shoot out of the back.

They trucked it to

Tesla Inc.’s

headquarters, in hopes Mr. Musk would accept the gift. The creators are calling called the event “GOATSgiving.”

But about two hours after the co-founders of Elon GOAT parked the sculpture right outside the Tesla building, there was no sign of Mr. Musk.

Dustin Dailey, a security officer at Tesla, walked over to a group of about 15 people and said they couldn’t accept the sculpture on Mr. Musk’s behalf, but would find a spot for it on their property if Mr. Musk gave the thumbs-up.

But so far Mr. Musk hasn’t given any indication he would accept it or whether he knew the sculpture was there. Tesla didn’t respond to a request for comment

“I am fairly certain he does know about it,” said Mr. Dailey of the sculpture. “It’s all over Twitter.”

Alec Wolvert, an Elon GOAT co-founder and chief marketing officer, said they were planning on camping out on a piece of public land off a toll road that overlooks the headquarters until Mr. Musk accepted the sculpture.

“We’re gonna stay here as long as possible,” Mr. Wolvert said. “I even heard some people say they were going to strap themselves to it.”

The idea of the sculpture came together last year. “It was an evening joke that kind of just came to fruition,” said

Ashley Sansalone,

an Elon GOAT co-founder.

The cryptocurrency entrepreneurs asked Kevin Stone, a metal sculptor in British Columbia, Canada, to make the giant sculpture with Mr. Musk’s head. The goal: to get Mr. Musk to tweet about the sculpture to his more than 118 million followers and draw attention to their cryptocurrency, the Elon GOAT.

“Elon tweeting us would legitimize the token,” said Mr. Sansalone, 40 years old.

Mr. Sansalone said he works on the token full time and previously ran a construction company and traded energy. Unlike bitcoin, ether or dogecoin, the Elon GOAT token is far from a household cryptocurrency name. It is ranked well outside the largest cryptocurrencies by market value, according to CoinMarketCap.

Mr. Musk’s head, which took nearly six months to complete was made by Mr. Stone. The goat body and rocket were made by others in Phoenix to speed up the project, Mr. Sansalone said. Then all the pieces were put together and attached to the back of a 70-foot long semi-truck trailer.

“When I first saw the statue my jaw dropped,” said DeMarco Hill, 51, who spotted it in September in Goodyear, Ariz., where he lives. He grabbed his 12-year-old son and they followed it. “It was something you’ve never seen before in your life.”

Mr. Hill, a trucker who owns his own company, Stay Ready Trucking, thought the stunt was so entertaining that he found Mr. Sansalone and asked if he could participate. Mr. Sansalone said Mr. Hill was needed because only someone with a special license could drive around the heaping pile of metal.

He has since driven the sculpture through California, Arizona and Washington, before bringing it to Texas. People who drive by honk their horns or give a thumbs-up, Mr. Hill said. 

“If I pull up to the side of the road it’s like people crowding around,” he said. “It gets crazy.”

Mr. Sansalone said the sculpture has mostly gotten a positive response. He hasn’t heard anyone mistaken Mr. Musk’s face for someone else. “I would say he is probably the most relevant person on the planet right now,” Mr. Sansalone said about Mr. Musk, the world’s richest person who recently bought Twitter Inc. for $44 billion.

In September, the sculpture sat in front of Tesla’s office in Palo Alto, Calif., during the company’s artificial-intelligence conference. Tesla employees crossed the street to take pictures with the sculpture, Mr. Sansalone said. Mr. Musk was at the conference, according to Twitter posts he made, and Mr. Sansalone assumes the billionaire saw the sculpture. 

“All there was to look at was a lit-up rocket erected in the middle of the street,” he said. 

On Saturday night, the group remained hopeful.

At one point in the evening, a group of about 20 people who were waiting outside started to chant “Elon claim your goat” in the hopes that the god of crypto, as one co-founder put it, would hear them.

“I’m a huge fan of Elon and I want to give this man his flowers while he’s alive,” said Aamir Manzoor, a 36-year-old from Toronto who is a holder of Elon GOAT. “He’s done a lot for the world.”

Write to Joseph Pisani at joseph.pisani@wsj.com, Alyssa Lukpat at alyssa.lukpat@wsj.com and Adolfo Flores at adolfo.flores@wsj.com