RE: What AJAX framework do you recommend for PHP applications?
The choice of an AJAX framework has more to do with the application at hand than the backend platform. AJAX is purely a clientside technology and the various libraries and even platforms (Flex for one) communicate with the backends using text based protocols; JSON and XML are the most popular. All backends speak JSON and XML. All of them. However architecture is very different. If you are going truly AJAX the Controller and View layers could be very decoupled from the Model... It could live in Javascipt.
If you are building a data rich business front end look at ExtJs or Flex (not specifically AJAX but very powerful) an up and coming library is SproutCore (Apple). Prototype is an effects library and will do an admirable job of handling the communication and providing a solid foundation for your own dhtml interface.
On the back end php or otherwise there are security considerations to be handled (every single call has to be both secure AND atomic). Scalability can also come into play as authorization and security considerations can cause heavy load. Think REST and MVC.
Basically there is no simple answer to this question but I would spend more time matching the library to the application than php.
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