As mentioned earlier, I thought it might be a good idea to try the taylor series approach. However unfortunately this quickly breaks down, I tried to approach it my doing this:
fn(x) = a0 + a1x + ... + anxn
Then we have:
fn(fn) = g0(a0, a1, ... , an) + g1(a0, a1, ... , an)x + ... + gn-1(a0...