There is (in a perfect inductor) no resistance, so any finite voltage difference would eventually cause infinite current to flow (I=V/R) after an infinite time.

But as in the case of most divide by zero situations, you need to look at the the problem a different way to see what really happens.

If you suddenly apply a DC voltage to an inductor, the current rises linearly over time at a rate controlled by the value of the inductance, and the magnitude of the voltage.

One way to visualize what limits the rate of increase is to think of the inductor as its physical analog, which is an object with inertia.

Picture a heavy flywheel. Picture pushing on the edge of that flywheel with a constant force. The inertia of the flywheel pushes back with exactly the force applied (if it didn’t, the object pushing on it would slip relative to the flywheel.)

But the flywheel does not instantly start spinning - it gradually speeds up as its inertia is overcome by the push that is trying to make it move. If the force applied is constant, the rpm of the wheel will increase linearly over time, and the energy stored in the flywheel will also increase.

And inductor’s behaviour is quite closely analogous. The current corresponds to the RPM, and the energy stored in the growing magnetic field corresponds to the energy stored in the inertia of the wheel.