[Ichneumon]

Which was my whole point. Universal Turing Machines can't be produced either yet most of theoretical computer science is predicated on it. Many well-educated but otherwise foolish people forget that UTMs don't exist and are qualitatively different than what we can produce.

Well... The processing mechanism is straightforward to produce, but that infinite tape is the real b**ch.

But for a real tour de force, check out this Turing Machine built using Conway's game of Life. (Short background on Conway's game of Life can be found here.) Who would have believed that a grid operating under such a simple ruleset (originally conceived as just an idle amusement) would contain enough richness of behavior to enable the existence and operation of a full Universal Turing Machine? And who would have thought that a handful of subatomic particles would contain enough richness of behavior as to enable the existence of biological life?

[Doctor Stochastic]

Wolfram's rule 110 also produces a UTC. This rule is almost as simple as Life.