When you're defining your own classes, you should at a minimum inherit from NSObject.
If you want to define a custom button for use in an iOS app, for example, and the provided UIButton class doesn't offer enough customizable attributes to satisfy your needs, it makes more sense to create a new class inheriting from UIButton than from NSObject. If you simply inherited from NSObject, you'd need to duplicate all the complex visual interactions and communication defined by the UIButton class just to make your button behave in the way expected by the user. Furthermore, by inheriting from UIButton, your subclass automatically gains any future enhancements or bug fixes that might be applied to the internal UIButton behavior.
The UIButton class itself is defined to inherit from UIControl, which describes basic behavior common to all user interface controls on iOS. The UIControl class in turn inherits from UIView, giving it functionality common to objects that are displayed on screen. UIView inherits from UIResponder, allowing it to respond to user input such as taps, gestures or shakes. Finally, at the root of the tree, UIResponder inherits from NSObject.