How to fix a field with unlimited values where the "Add another item" is broken?
After many different tests, I have found the solution.

All the multi-value core fields using the module "Better Field Descriptions" were broken.

The help text added below by this module seemed to block the new option line to be created or at least displayed. Deactivate the module fixed the problem.

So, if you need a multiple values field and a permission to edit the help texts, be cautious with module interactions.

Thanks for those who tried to help me on this issue.