Every week, about two dozen patients come to a small room in Frederiksberg Hospital, a maze of old red-brick buildings in central Copenhagen. They are blindfolded and told to insert earphones with music. Then a nurse injects them with what they hope is the blockbuster weight-loss drug Wegovy.

The patients are trying to shed an addiction, not unwanted pounds. They are volunteers in one of a growing number of studies begun in the U.S. and Denmark this year to see whether Wegovy can treat alcoholism.

Researchers are pushing the boundaries on a class of drugs that became famous for helping people lose weight, testing the shots across a spectrum of disorders from Alzheimer's disease to sleep apnea. The market for obesity and diabetes will probably soar above $140 billion by 2032, analysts at J.P. Morgan predict, with the potential for even more if the drugs prove medically versatile.