A superconducting wire has no electrical resistance and as such it does not heat up when current passes through it. Non-superconducting wires can be damaged by too much current, because they get too hot and melt. But since current is the flow of electrons, surely unlimited electrons can't flow?
There is a critical current density for every superconductor where the superconductor acts as an ordinary conductor and a voltage difference can be measured between its ends.
The problem with trying to pass huge amounts of current through a superconductor is that any flowing current creates a magnetic field circling around it (Ampere's law). A superconductor also expels all magnetic fields from inside itself (Meissner effect), so what you get is a lot of magnetic field lines all bunched up just outside the surface of the superconductor, which makes a strong magnetic field there.
Magnetic fields kill superconductivity. Every superconductor has a "critical field" at which it stops superconducting (which actually depends strongly on temperature). So at some point the magnetic field produced by the current will be too much for the superconductor to handle, and it will "quench", which means it abruptly stops superconducting, and then the energy in the flowing current is very rapidly turned into heat.
A superconducting magnet quench can produce a sizeable explosion!
In addition to the fundamental limits noted by others there is the practical engineering issue of dealing with the increasing heating at the point where the superconductor joins the normally conducting parts of your circuit.
Get it wrong, and "Hello, Quench City!"