Anyone who reaches for Ohm's law and Kirchoff's laws and saying voltage/zero = infinite current is in error! Those are low-frequency approximations. Things can happen only at the universal speed constant (speed-of-light). (It will be assumed that the original poster has not called 1-800-GOD-PHYSICS-REQUEST to have the speed of light limit removed.)
What would happen when an thick all-temperature superconducter wire is attached to a one-trillionth ohm internal resistance battery?
The wire has to be at some potential value. Maybe halfway between the potential values of the two battery terminals. Where the wire touches the (-) terminal, electrons in their random fuzzy wavefunction wanderings fall into the wire. More and more. As they move into the wire an electric potential builds, an electric field appears in addition to what was naturally around the battery.
At the other end, electrons fall out of the wire into the battery (-) terminal.
The leading edge of the crowd of electrons at the (-) end can propagate only at the speed of light at best. Same for the leading edge of the dearth of electrons at the other end. Moving regions of charge = moving electric fields = magnetic fields. Changing electric and magnetic fields = radiation.
For an "ideal" zero resistance wire, how do those regions of charge move, and how do new charge carriers enter in? If it runs away to infinity, then also an infinite about of radiation can be expected.
What happens when the crowd of extra electrons meets the dearth of electrons near the middle of the wire? Then, for regular devices, a steady current would be established. Where the cloud of extras meet the cloud of dearth, they cancel out. We have a growing region of normal electron density. More changes to a moving electric field, affecting radiation.
Soon that normal region reaches the battery terminals. Then - we have the original situation where we just touched the wires to the battery! Have we made some kinda oscillator? I doubt it. Electrons are quantum particles, and randomness plays a role, so the leading edges of the various regions aren't sharply defined but fuzzy and maybe get fuzzier. A proper treatment would require using equations for semi-classical electron transport (found in, for example, Ashcroft & Mermin) except, being laws of physics, they'd be hard to apply to a nonphysical system.
Another way to look at this is by inductance.
In this absurd example, the current is destined to reach a steady state of infinity, the magnetic fields are going to be, uh, rather strong. The inductance of the wire is something to think about. Inductance exists anywhere charge moves through space. "Ideal" wire isn't going to be lacking inductance. Current always takes time to get flowing, to reach its steady state value.
I'm not totally sure, but I suspect that the rapidly increasing magnetic field created by the increasing current offers resistance to the current. Remember, inductors 'like' to maintain steady current flow. The runaway current will increase linearly. The inductor equation
$$V_L(t) = L\frac{di(t) \over dt}$$\$V_L(t) = L\frac{di(t)}{dt}\$ is easily integrated to give
$$i(t)=\frac{ V \over L}t$$\$i(t)=\frac{ V \over L}{t}\$ where \$V_L(t)\$ has been set to the battery voltage \$V\$.
At some point, the magnetic field will be strong enough to twist Earth out of whack and we'll finally have winter in San Diego :) I like winter, so I'll fund your experiment...
Skin Effect
Another thing to think about is the skin effect. Currents like to squash themselves out to the surface of the conductor they flow through. At some point, the current will be so great and so squashed, that even in a superconductor or ideal conductor, the graininess of matter will put a stop to the runaway growth.
