What if we keep increasing the external electric field—as electrons in a conductor are finite–so that after saturation there won't be further electrons to nullify the external electric field? In this situation, will there be any field inside the conductor?
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$\begingroup$ Sure, once you don't have enough electrons to screen the field, there will be some field in the material. $\endgroup$– IcyOtterCommented Aug 16, 2021 at 2:44
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Conductivity ($\sigma$) is in units of $\frac{A}{ V}$ - in other words how much current do you generate per unit of potential difference across two points? So if you've gotten to a point where there are no free charges and no current results from changing the potential, you no longer have a conductor.
