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Ok, this is fascinating. I love cakephp and also love joomla. When I found out about this, I got very excited. Have not tested it yet, but I am going to very soon.

Jake is the child of joomla and cakephp.

Jake is a component that lets you execute your CakePHP applications inside Joomla. It is meant to provide a transparent bridge between Joomla and CakePHP, allowing your applications to be run with or without Joomla, with absolutely no need to change your existing code.
Jake is still in beta stages, but I am sure that we are going to have a stable release soon.

More info about it at Jake website.

There were times when I wanted to use Joomla! and integrate it with new functionalities. The best option would be to create new components for joomla, but I never got to do this, did not want to invest time in this. Now, that belongs to the past. I  just discovered this article by Max aka Abhimanyu Grover. He illustrates how to install joomla! within the cakephp framework and take all the advantages of cake while at the same time you get the user authentication and content management that joomla! offers. Great article, take a look.