Well to break it down:
1. create a 'round rectangle button' that when pushed, will create a 'text field'.
2. when the button is pressed, the new 'text field' needs to orient with the button, and the button needs to move down out of the way so that the two objects do not overlap
3. the effective dimensions of the scroll view need to expand every time the button is pressed to make room for more text fields.
Take the analysis further, and what do you have?
After pressing the button 15 times, you'll have a
table of 15 text fields arranged in a
single column, with a
button below them.
What would you normally use to display a table? UITableView.
So read about that class, and its data source, delegate, and cell relationships. It scrolls. It automatically expands when you insert a cell. It moves cells up or down depending on where you insert the cell. Other than the button below, a UITableView of text fields would be exactly what you want.
So now think about different ways you can make the item below the last text-field a button. Can you place a sub-view there containing the button? Can you make the last table-cell a different cell-type that contains a button? Read the reference docs.