I tried to play the free roll without solving the captcha to see how it works.
For me, the cost to do that for the third time wasn't 2^2 reward points. It was 3 reward points. So, I guess it would increase by 1 every time you play the free roll without solving the captcha.
I tried it today:
1, 3, 4, 7, 11 and guess that next will be 18 because it was a
Fibonacci sequence.