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.