If the information for each “add more” section can fit on one line (in several columns across) you can use the List field:
If you need to collect more information than that, Gravity Forms has a Repeater functionality, which requires code to expose:
If you want a solution that does not require any code, you could use this:
Finally, if you only need to allow something like 4 possible ‘sections’, you could always add all the fields to the form, within section breaks, and use conditional logic to show another section when they click an “add more” button. This would not require any add-ons, but you would need to know the maximum number of sections you are going to allow, because they would all need to be added to the form when you build it, and each new section is only shown if they click an ‘add more’ button.
If you need any more information, please let us know.