The way I understand it, electric wires heat up because there is too much current passing through them. There are tables for various materials, how much current can pass through a given cross-section without a significant heat-up. That's also why we use high voltage in the big power lines to transfer a lot of power - this allows us to make the wires thinner (cheaper).
But is there an upper limit to this trick? Would it be (theoretically) possible to raise the voltage to petavolts, and pass all the power of the entire planet through a wire no thicker than a hair?
Added: To clarify, I'm wondering how far the voltage can be pushed, what the problems would be (ok, so insulation is the first one), and if there are any theoretical limits that cannot be surpassed no matter how much technology improves.