One crude option is the make multiple layouts with an increasing number of fields so you can define the layout of these fields. You can set-up your button to move to the layout with more fields or have multiple buttons that go to layouts with the desired number of fields.
FileMaker cannot dynamically add fields to a layout nor to a table.
This sounds like a case where you need a table of related records, one record for each of these fields. Then you can add as many of these fields as you need by adding related records in one or more portals.
Okay, I knew it would probably be a stretch.
Thanks for your answers though. Much appreciated!