What is the optimal way to perform GCD chain operation?
You are saying that you will perform n-1 gcd operations, unless you find k numbers with a gcd of 1 by performing k-1 gcd operations. 

Since you don't know which set of numbers has a gcd of 1, you can't pick a set with gcd 1. Your total time depends on the data you have. 

You can try to process small numbers first, because the gcd of small numbers tends to be calculated quicker than the gcd of large numbers. 

The optimal strategy will not always lead to the best outcome. I put a deck of cards on the table, face down, and ask you to pick the ace of hearts. Do you think "pick the ace of hearts" is the optimal strategy? No, it isn't a strategy that you can enforce since you don't know which one is the ace of hearts.