In-N-Out's popular burger style started with a customer request decades ago

In-N-Out Burger is known for its simple menu, with basic burgers, fries and shakes. Yet by 2005, the fast-food chain was listing some of its most popular off-menu orders on a "Not-So-Secret Menu" online.

Among the most famous off-menu offerings is the Animal Style burger, which the company website says is a "burger of your choice with hand-leafed lettuce, tomato, a mustard-cooked beef patty; add pickle, extra spread, with grilled onions."

In-N-Out’s website says the restaurants accommodate...