Jeremy Bowers is kind enough to answer my question regarding why you can't have an only almost general purpose computer.
As much fun as this explanation, the proof of this is one of the most difficult and twisty proofs in the history of mankind (in my opinion), because of the way the Universal Turing Machine wraps back on itself to run itself. So that's virtually useless in any general description. Moreover, any such demonstration must necessarily get down to the nuts and bolts of what computability is, which means tossing around some very ugly-looking mathematical notation.
