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super() is a special use of the super keyword where you call a parameterless parent constructor. In general, the super keyword can be used to call overridden methods, access hidden fields or invoke a superclass's constructor.
The benefits of super() in single-inheritance are minimal -- mostly, you don't have to hard-code the name of the base class into every method that uses its parent methods. However, it's almost impossible to use multiple-inheritance without super(). This includes common idioms like mixins, interfaces, abstract classes, etc. This extends to code that later extends yours. If somebody later wanted ...
In fact, multiple inheritance is the only case where super() is of any use. I would not recommend using it with classes using linear inheritance, where it's just useless overhead.
Just a heads up... with Python 2.7, and I believe ever since super() was introduced in version 2.2, you can only call super() if one of the parents inherit from a class that eventually inherits object (new-style classes).
Thirdly, when you call super() you do not need to specify what the super is, as that is inherent in the class definition for Child. Below is a fixed version of your code which should perform as you expect.
As for chaining super::super, as I mentionned in the question, I have still to find an interesting use to that. For now, I only see it as a hack, but it was worth mentioning, if only for the differences with Java (where you can't chain "super").