Is there an upper limit of voltage than can pass through a wire?
You can keep increasing the voltage on a wire up to a point. Unfortunately higher voltage does require thicker wire, and it's nothing to do with heating.

The electric field at the surface of a single wire of given radius is proportional to the voltage/radius. Once the electric field exceeds the breakdown field of the dielectric that's insulating the wire, it begins to break down the insulation. This creates corona if the dielectric is a gas, like air of SF6, and damages the insulation if it's a solid.

This is handled on overhead wires by making a bundle of thin wires. This behaves, as far as corona is concerned, as a much thicker wire. Each wire is 'shielded' to some extent by the others, which reduces the electric field at its surface. While the 275kV grid tends to use a bundle of four wires, MV grids use several 10s of wires and a bundle more than 1m in diameter.

Many of us have seen pictures of high voltage switchgear. The conductors between nodes are often very large diameter tubes. This is simply to allow air insulation to be used with minimal corona