The solution, for any length of chain, is going to be equivalent up to scaling... that is, if you double the size of your initial circle, you just double the size of all the other circles, and your final distance is also doubled. Which means we can solve this recursively... if we know the ideal sizes for an n-1 chain, we can treat that as a unit and figure out how to scale it and position it on another circle, and end up with the ideal n chain.
The base case is the 1 chain, which obviously has a length of 1.
Now, say the n-1 chain has a length of L. To make an n chain, we have a unit circle, and then at a distance of x from that, we have the n-1 chain, and we need to find x. Now, the n-1 chain will be scaled down by sqrt(1-x2) to satisfy the chord-diameter thing, so the full length will be x + L*sqrt(1-x2). A quick differentiation finds that this is maximised when x = sqrt(1/(L2 + 1)). The n-1 chain will then be scaled down by a factor of sqrt(L2/(L2 + 1)). Substituting this back into our formula for the full length then gives sqrt(1/(L2+1)) + L*sqrt(L2/(L2+1)), which simplifies to just sqrt(L2+1). Given how simple this result is, there has to be a simpler way to derive it, but whatever, this works.
So, working from here, the maximal length of a 2 chain will be sqrt(2), the maximal length of a 3 chain will be sqrt(3), the maximal length of a 4 chain will be sqrt(4), etc. So the maximal length of the 16-chain will be sqrt(16) = 4.