What is the purpose of python's inner classes?

Quoted from http://www.geekinterview.com/question_details/64739:

Advantages of inner class:

• Logical grouping of classes: If a class is useful to only one other class then it is logical to embed it in that class and keep the two together. Nesting such "helper classes" makes their package more streamlined.
• Increased encapsulation: Consider two top-level classes A and B where B needs access to members of A that would otherwise be declared private. By hiding class B within class A A's members can be declared private and B can access them. In addition B itself can be hidden from the outside world.
• More readable, maintainable code: Nesting small classes within top-level classes places the code closer to where it is used.

The main advantage is organization. Anything that can be accomplished with inner classes can be accomplished without them.

Is there something that can't be accomplished without them?

No. They are absolutely equivalent to defining the class normally at top level, and then copying a reference to it into the outer class.

I don't think there's any special reason nested classes are ‘allowed’, other than it makes no particular sense to explicitly ‘disallow’ them either.

If you're looking for a class that exists within the lifecycle of the outer/owner object, and always has a reference to an instance of the outer class — inner classes as Java does it – then Python's nested classes are not that thing. But you can hack up something like that thing:

import weakref, new

class innerclass(object):
    """Descriptor for making inner classes.

    Adds a property 'owner' to the inner class, pointing to the outer
    owner instance.
    """

    # Use a weakref dict to memoise previous results so that
    # instance.Inner() always returns the same inner classobj.
    #
    def __init__(self, inner):
        self.inner= inner
        self.instances= weakref.WeakKeyDictionary()

    # Not thread-safe - consider adding a lock.
    #
    def __get__(self, instance, _):
        if instance is None:
            return self.inner
        if instance not in self.instances:
            self.instances[instance]= new.classobj(
                self.inner.__name__, (self.inner,), {'owner': instance}
            )
        return self.instances[instance]


# Using an inner class
#
class Outer(object):
    @innerclass
    class Inner(object):
        def __repr__(self):
            return '<%s.%s inner object of %r>' % (
                self.owner.__class__.__name__,
                self.__class__.__name__,
                self.owner
            )

>>> o1= Outer()
>>> o2= Outer()
>>> i1= o1.Inner()
>>> i1
<Outer.Inner inner object of <__main__.Outer object at 0x7fb2cd62de90>>
>>> isinstance(i1, Outer.Inner)
True
>>> isinstance(i1, o1.Inner)
True
>>> isinstance(i1, o2.Inner)
False

(This uses class decorators, which are new in Python 2.6 and 3.0. Otherwise you'd have to say “Inner= innerclass(Inner)” after the class definition.)

There's something you need to wrap your head around to be able to understand this. In most languages, class definitions are directives to the compiler. That is, the class is created before the program is ever run. In python, all statements are executable. That means that this statement:

class foo(object):
    pass

is a statement that is executed at runtime just like this one:

x = y + z

This means that not only can you create classes within other classes, you can create classes anywhere you want to. Consider this code:

def foo():
    class bar(object):
        ...
    z = bar()

Thus, the idea of an "inner class" isn't really a language construct; it's a programmer construct. Guido has a very good summary of how this came about here. But essentially, the basic idea is this simplifies the language's grammar.