Question: This is the first time for me asking a question. Hope you'll be able to help me.

Problem: my chef wants our enterprise application to have a module which allows customers to upload very large files to our server.

Infrastructure: php 5.x / mysql client-server app

Well known problems:
-HTTP session time out
-Upload limit in terms of number of files and file size

Rules:

-The solution mustn't use any applet (java, flash, ...), neither any browser plugin.
-The solution should allow users to upload any type of files, from simple images to very large files of any other type.
-For security reasons, we don't want to change the settings in the php.ini file, neither in any .htaccess file to allow the app uploading larger files.
-If a pre-backed solution exists, it should be opensource and possibly free of charges.
-Integration with AJAX functionalities and progress bar visualisation are heavily welcome.

Possible solutions (tell me if it's possible, and if it isn't, why?):

-Access by FTP using some javascript library in the browser
-Access by WebDAV using some javascript library in the browser

**Any other solutions that respect the above given rules is also welcome** 

I know, I'm asking for a very hard thing to find.

Answer: 
What you're asking for isn't currently possible. The only technology for uploading files which is currently available in all widely used browsers is standard HTTP form uploads, which you've already rejected as a solution. All other solutions available require Java or Flash, or use browser JavaScript capabilities (local file access) which are not fully standardized, nor universally available.

You'll need to relax some of your requirements.
Create an iframe or frame, or browser-window that offers access to the server via FTP. Allows users to drop files there for those browsers at least who support FTP.