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Tom_McCabe comments on The Crackpot Offer - Less Wrong
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"I challenge the "rules" set out by whomever thinks he's the know-all on what can be done with a compass and straight edge."
I would be interested to see what you can get out of a compass and straightedge if you change the allowable operations. You could wind up with something much more complex than the things the ancient Greeks studied (think of how much more complex a Riemannian manifold is than a Euclidean n-space, once you remove a few of Euclid's axioms).
I know this is an old comment, but the answer is actually quite nice.
What the compass and straight-edge basically give you is the capacity for solving quadratic equations. There's a field of numbers between the rational and real numbers called the Constructible numbers that completely characterizes what can be done there.
Alternative techniques (e.g., folding) can allow one to solve cubic equations, and so the field of numbers that can be constructed in this way is an extension of the Constructible numbers.
So the full answer to "what you can get if you change the allowable operations" is that construction techniques correspond to field extensions of the rational numbers, and this characterizes their expressive power.
You are more than a paper-machine, you are a paper-based math expert.