Inverse Kinematics with more than 2 segments
What you have here is an underconstrained problem*. To solve this problem you do what you do to any other underconstrained system. This is also why you mostly only see the 2 joint version as it is not underconstrained in the planar solution, and in any case the preregistration is trivial.

Basically you have 2 options:


Add more constraints
Deal with the freedom


Adding more constraints to the system reduces the degrees of freedom. For example if you only permit one of the joints to rotate about its z axis and the another around its x axis then y then you have reduced the problem to the same 2 member problem. Another way to constrain it is to force the orientation of the last link. This also reduces it to the same case as 2 links. But there are other meaningful ways to reduce complexity that depend on your application.



Image 1: Constraining one member reduces the problem to a 2 member problem. In the planar case that means 2 solutions (other in blue).

Dealing with the freedom just means that your answer is going to be a family of answers. Then you develop a way to choose the solution from that family. Letting the user adjust the parameter is equivalent to adding one more constraint. But there are other methods such as least squares, or energy methods that yield different results. Sky is the limit here. For example I have worked with robot controllers that decided the parameter based on what was the least amount of rotation on joints. The problem was that this wasn't entirely deterministic so over time the robot ended up doing things differently...

* This is more suited to Engineering.SE but since IK is a common graphics problem lets deal with it here.