How can the Fermi level be in between the band-gaps?

Question

This isn't a duplicate of What does Fermi level in the band gap mean?. I read it and I'm still confused, and moreover I'm not allowed to add another comment to it anyway because I don't have enough reputation.

In particular:

the Fermi level $E_F$ is the chemical potential at 0 Kelvin.
chemical potential is the change in energy per particle added
the zero of the potential is at the top of the valence band

If I have a full valence band and I add another electron, I need at least the energy of the band-gap to promote it into the conduction band. So surely, that must mean $E_F$ can't be smaller than the band-gap.

How then, is the Fermi level ever inside the band-gap? I feel as though I'm missing something incredibly obvious here.

Possible duplicate of What does Fermi level in the band gap mean? — sammy gerbil, Jan 24 '17 at 23:35
The Fermi level energy is a parameter to a distribution/probability function (the Fermi function). By itself it is meaningless. The fact that the function spans a band gap does not mean much either. Could you clarify why you think the Fermi level cannot be inside a band gap? — Jon Custer, Jan 24 '17 at 23:44
The linked post says the Fermi level is the chemical potential of the electron. The chemical potential is the change in energy when an electron is added to the system, so if I have a full valence band I need at least the energy of the band gap to add another electron, so the Fermi level can't be inside the band gap. — dain, Jan 24 '17 at 23:56
This question is mentioned in a Meta discussion about commenting. — rob, Jan 25 '17 at 01:00
@dain, welcome to Physics.SE. I think that as long as you are clear about why the other answers don't help you, a new question will probably remain open and get answers. I've removed some comments on that non-physics topic. — rob, Jan 25 '17 at 01:12

How can the Fermi level be in between the band-gaps?

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