The oldest accounts of bakers suggest that they kept a small piece of dough aside each day, and used that to leaven the next day's baking. This would've resulted in quite slow-rising dough, and bread that was probably fairly dense. It seems to have been the method of ancient Egypt, and perhaps ancient Semetic peoples and Greeks.

As /u/agentdcf said they made sourdough bread. Natural yeast lived in the dough, which needed to be continually 'fed' day after day with new flour and water.

Sometimes they found other methods of making leavened bread - Pliny the Elder in his Natural History says the Gauls used beer foam.

Sugar isn't necessary to make sourdough bread, but yeast can thrive on sugars, which may be why Pliny says grape mash/juice (passum) was used in the famous Roman Picenum bread. This bread was allowed to rise for nine days to get the natural yeast established. It was described as a 'light sponge' by Martial, so it definitely poofed up from natural yeast.

Of course, early bakers could also make unleavened bread, or bread which didn't rise.