Smallest possible resistance (highest current)
By Ohm's Law on it's own the current, theoretically, can go to infinity in a circuit with zero resistance. Of course, in reality you would need a superconducting loop to do that.

However, if you have an infinite current in a loop, you also generate an infinite magnetic field.

Once that field gets large enough, the super-conductor will cease to be super-conductive in an event known as quenching. Then things go BOOM!

Further, even if you can overcome the above, you would run into quantum and relativistic effects.