"... suppose we define f(X) over integers as taking the 1st digit from the left, then the 2nd, then the 4th, then the 8th, etc, so f(12345)=124 and f(12345678)=1248
Set y=f(f(f(f(f(…f(Graham’s number))))) such that there are the fewest number of fs such that y has less than a million digits." 
y is Avi's number. Is Avi's number a prime or composite number?
Because we know the last digits of Graham's number, it seems to me that there might be a way to determine if the last digits of Avi's number is definitely not a prime.