A PICAXE is designed to do a lot as is. But making a full computer out of a PICAXE is like turning two motorcycles into an automobile. Sure you can bolt them together with a frame and add a shell and windows around them and spend months making it all work together, but why not just buy a car?
The PICAXE is much like a motorcycle. It comes complete with everything needed to do specific jobs, but that attribute makes it awkward for it to be used as a generalized computer.