South American Lungfish, African Lungfish, Arapaima and Electric Eels are all obligate air breathers and can drown without access to surface air. Interestingly, the Lungfish and Arapaima have fully functional gills when babies and transition to air breathing when they get older.
Some anabantoids are said to be obligate air breathers, but I have not been able to find any reliable sources for this claim. It was often said that Bettas, when shipped in breather bags, needed an air pocket in the bag. I tested this claim by placing 10 Bettas in breather bags, 5 with air pockets and 5 completely fillled with water. After one week, all 10 Bettas were still alive, so they cannot be obligate air breathers.