At what resistance can a real inductor act like an ideal inductor?
But is there a low resistance at which point the current would increase at a constant rate like an ideal inductor?

Sure that point is where the inductor is ideal meaning that the series resistance is zero.

So you could say that in the real world (where there are no ideal inductors only ones that have some series resistance, also assuming no tricks such as superconductivity at low temperatures) there is no inductor which behaves as an ideal inductor. And you would be correct.

The thing is, how significant is the difference caused by the series resistance. Let's assume that the current decreases 0.1% because of the series resistance then you already need a pretty accurate current measuring setup to be able to measure that change.

In electronics many things are not ideal. Even a 0 ohm resistor has a series inductance of a few nH. Does that affect the behavior of all circuits? No only some RF (high frequency) circuits. RF designers know this and take the that (parasitic) inductance into account. In non-RF circuits the small inductance of a resistor is insignificant.

Same with inductors. A designer using an inductor will take into account that it has some series resistance. If the series resistance is too high then maybe the designer selected the wrong inductor. A designer should choose the right inductor such that the series resistance become insignificant.