PeopleExpress used to do this. When it first launched, customers purchased tickets for a flight without booking a particular flight or seat. They would show up to the airport and get seated on a first-come, first-serve basis on the next available flight. All flights were non-stop shuttle flights (back and forth between two destinations).

It was phenomenally successful for a few years. It's not clear that this model could be replicated today due to air travel security concerns.