Rinse and core apples; discarding seeds. Cut apples into 1- to 2-inch pieces. Transfer apple pieces to a large, heavy pot. Add water. Cover and cook over medium-high heat until the apples start to break down, about 10 minutes.
Stir apples. Reduce heat to medium, and cook 15 minutes. Reduce the heat to low. Simmer, covered, 30 minutes more, stirring often so that the applesauce does not stick to the bottom of the pan. Check the apples with the tip of a knife for softness. If still slightly firm, continue to cook, with lid ajar, until apples are broken down and slightly chunky, 10 to 20 minutes more. Applesauce will be chunky at this stage. For smooth applesauce, let cool and process in batches in a food processor. Let cool; refrigerate up to 3 days in the refrigerator or freeze for up to 1 month.