At first, residents of the upscale cul-de-sac in West Lake Hills, Texas, did not know who had moved into the 6,900-square-foot, six-bedroom mansion next door.

Then construction workers arrived to erect a 16-foot chain-link fence around the $6 million property, which is one of four homes on the leafy street. They also installed an outward-facing camera. Next, a fleet of cars — many of them Teslas — began parking on the street. Three times a day, a shift change signaled security personnel coming and going at the house. Once, the driver of a passing car shouted late at night that he was looking for a party at “E’s house.”

No one liked the commotion, or the traffic, or the keypad-activated gate opening and closing for workers and cars at all hours. So even when they learned through word of mouth that their new neighbor was Elon Musk, the world’s richest man, that did not stop some of them from grousing to the City of West Lake Hills about his mansion.

The complaints have since escalated into an uproar over city ordinances, permits and exceptions known as variances — so much so that the matter of Mr. Musk’s house landed in a contentious Zoning and Planning Commission meeting last month. The debate is headed next to a West Lake Hills City Council session, scheduled for May 14.

“Transporting service employees to other houses, leaving their cars on our quiet streets, hauling laundry to and fro to other houses has to stop,” Paul Hemmer, a neighbor and the main complainant, wrote to the Zoning and Planning Commission. The letter was also signed by the occupants of the two other houses on the street.

For the past few months, Mr. Musk, 53, has bulldozed his way across Washington, flouting long-established traditions and hacking away at what he deemed to be unnecessary federal bureaucracy. But in his own backyard outside Austin, the tech billionaire has become mired in a maze of local regulations and red tape. No one, it seems, is rich enough to escape the neighbors.