JAKARTA, Indonesia — On a recent morning driving his motorbike for one of Asia’s fastest-growing tech start-ups, Nasrun picked up and dropped off four schoolchildren, an office worker, medicine from a pharmacy, some dumplings with peanut sauce, a few documents and an order of Japanese food, the last of which he took to a woman at the Indonesia Stock Exchange.

For his colleague Irawan, the workday started at midnight. He ferried someone home, then delivered an order of KFC. Around 4 a.m., he picked up a clubgoer so wobbly with drink that Mr. Irawan had to use one hand to steer and the other to keep the woman from falling off the back of his motorbike.

The two men work for Go-Jek, a $3 billion Indonesian start-up whose maximalist approach to the ride-hailing business has put rivals like Uber on notice, and gotten the attention of American investors and Chinese internet titans alike.

The company’s main app lets you summon a car or motorbike driver who could just give you a lift, sure — but who could also bring you takeout, shop for groceries or deliver a present to someone across town.