Flatten double-linked list
Your current implementation is quite nice. 

The only way you are going to avoid using the while loop to find the right-most node is if you have each individual branch maintain its branch's tail Node. This is a [sort of] good approach to the problem. However, once you reach this point, a Node really starts becoming more than just a Node: it now has information about the entire structure instead of what is directly around it.

So instead of modifying the Node class, you could create a Tree class. 

This change makes sense: 


Right now you manually create your 'tree'. This approach is not very portable. A Tree class can handle its own creation.
You need to store information about the entire branch structure somewhere. Inside a Node doesn't feel right.
You don't really flatten a Node, you flatten a tree of nodes. Thus, it again makes sense to move the flatten function inside a Tree class (or at least take a Tree parameter).


Here is a skeleton version of my Tree class. I have stubbed out all my methods except flatten as that is the function in question:

class Tree(object):
    def __init__(self, root=None):
        self.root = root
        self.tail = root        
        self.length = 1 if root else 0

    def __iter__(self):
        ''' Allows iteration through the tree using `for node in tree`. '''
        pass

    def __getitem__(self, index):
        ''' Fetches the `Node` at the passed index. '''
        pass

    def grow(self, nodes):
        ''' Appends node/nodes/Tree onto itself. Be careful when adding `Node`
            we need to append a *copy* of a `Node`, not the reference. '''
        pass

    def graft(self, tree, node_pos=0):
        ''' Grafts in another tree to this one. By default the new tree
            is grafted to the root of the tree. '''
        pass

    def flatten(self):
        ''' Returns the flattened version of itself. '''
        flat = Tree()
        for next in self:
            flat.grow(next)
            if next.child:
                flat.grow(next.child.flatten())
        return flat


In this version of flatten, we keep the recursive feel as we call a method with the same method declaration. However, we are not doing true recursion because the methods are bound to different objects, and thus are different methods. 

In any case, the flatten function is really quite simple:


Create the new Tree we are going to modify and return.
Loop through the Nodes of the current tree.
Append the current Node.

3.1. If that Node has a child Tree, append the flattened version of that Tree.
Return the final flattened Tree.


Here is my Tree class plus your Node class in action:

one, two, three, four, five, six, seven, eight, nine = [Node(num) for num in xrange(1,10)]

tree = Tree(one)
tree.grow([two, three, five])

# Graft the new `Tree` to the 3rd `Node`.
tree.graft(Tree(eight), 3)

branch = Tree(six)
branch.grow(seven)

# Graft the new `Tree` to the 1st `Node`.
branch.graft(Tree(nine), 1)

tree.graft(branch, 1)

def flatten(tree):
    ''' A non-class version of the `flatten` function. '''
    flat = Tree()
    for next in tree:
        flat.grow(next)
        if next.child:
            flat.grow(flatten(next.child))
    return flat


print ' - '.join([str(node.value) for node in flatten(tree)])
print ' - '.join([str(node.value) for node in tree.flatten()])