Is there a way to have a field created on a form based on completion of another field or use of a button

OK, so you don't want to create new fields on the fly, just expose them.

No problem

So you can have a form rule that says
When approval 1 is not blank, show approval 2

Then another for rule that says
When approval 2 is not blank, show approval 3

etc up to 11 rules in total