Series where each term is square of previous The problem statement is as follows:-


  In a court case, a judge cited a court of contempt and ordered a fine of $2 for first day.
  On subsequent days, the fine would be equal to the square of the previous day's fine.
  e.g. 2, 4, 16, 256 and so on.
  Find out:-
  
  a) What would be the fine on day $n$?
  
  b) How many days would it take the fine to reach $D$ dollars?


A big Oh answer would be appreciated.
As you say, the fine on day $n$ is $2^{2^n-1}$. To reach $D$, you need to solve the equation $2^{2^n-1}=D$, taking $\log$ we have $2^n-1=\log D$, rearranging and taking $\log$ again we get $n=\log(\log D+1)$.

If you want the answers in Big-O notation, you have $2^{2^n-1} =O(2^{2^n})$, and for the second part, $n=O(\log\log D)$.