More than 1,000 times a minute, someone bites into what has been described as a wet envelope of cat food—and keeps eating.

Jack in the Box is known to most of the country for its hamburgers and bigheaded mascot. But for many of its devotees, the magic of the fast-food chain lies in its interpretation of a taco.

A tortilla wrapped around a beef filling that is dunked in a fryer and topped with American cheese, lettuce and hot sauce, the taco appeared on the menu in the 1950s after the first Jack in the Box opened in San Diego. As the chain spread beyond California, the taco has followed it —with good reason. Jack in the Box now sells more tacos than any other item on its menu thanks to a legion of fans who swear by the greasy vessels even as they sometimes struggle to understand their appeal.

The first time Heather Johnson tasted a Jack in the Box taco, she was at a drive-through in Cincinnati when she noticed you could get two for 99 cents, so she added them to her burger order.

She took two bites, threw the rest on the passenger seat and kept driving. “It was stale, greasy, spicy, crunchy, saucy and just plain strange,” said Ms. Johnson, a 43-year-old director of operations at an advertising agency in Cincinnati and author of a blog called the Food Hussy. “Who puts a slice of American cheese in a taco?”