Complexity is O(log(log(n))) but how can i prove that?

Question

I'm new at the topic of complexity analysis and I have a while loop which stars from 3 and goes to n. It grows exponentially anyway but i'm quite bit confused when comes to prove this mathematically. How can i show step step values of i in terms of kth step. What is mathematical description of big-Oh of this loop ?

while i < n
    i = i^2 -i + 1

Answer 1

As the numbers grow, they approximately square at each step, so the growth is doubly exponential.