I bought four cows and didn't name one of them. The one I didn't name disappeared after one day. (Tater Update: Can't have no name for animal, tried using just a space as the name, filling the name box then deleting, and typing one character and backspacing and pressing confirm at the same time and none worked. No name animals called " " did not despawn. Probably fixed with recent animal purchase fixes, but can't reproduce to say for sure.)