You referring to Maxwell's equations yet again tells me that you have not read the answer!

@AlfredCentauri Sigh. How many times do I have to repeat that "The chemical forces cannot be explained by classical electrostatics, which includes Maxwell's equations"?

I am not sure how else to convince you. As far as I know, there are quantum-mechanical effects involved, and Ben Crowell has explained them in his answer.

You can, by all means, continue believing that there has to be a time-varying magnetic field in order to have an emf, and that a conservative electric field can enable a battery to drive charge repeatedly along a loop.

Let me quote this again: "aren't [Maxwell's equations] universally valid, compatible with both special relativity and quantum mechanics? No, they're not. Maxwell's equations are classical. For example, you're not going to be able to explain the photoelectric effect using Maxwell's equations".

Once again, you're avoided answering the simple question I've asked. Ben's answer is clear as is the Wikipedia article. The emf due to chemical and thermal forces does imply that the electric field is non-conservative (look closely at the three integrals for the emf at the Wiki article). You've stated that the electric field of the battery is non-conservative but, if that were so, there must be a time dependent magnetic field.

In summary, you seem to saying that a non-zero emf demands a non-conservative electric field despite Ben's answer and the Wiki article.

@AlfredCentauri The answer to your question is "There DOES NOT need to be a time-varying magnetic field to have a non-zero curl". In how many ways must I tell you this? In Japanese maybe?

There is NO magnetic field in any way.

Also, don't hijack Ben's answer. Nowhere in his answer does he say that the field is conservative. The electric field is defined as the force per unit charge. If there's an emf driving the charge around, it must be part of the field by definition.

"In summary, you seem to saying that a non-zero emf demands a non-conservative electric field". Yes, this is correct. $IR = V + \mathcal{E}$. The emf is part of the electric field as well. You seem to be saying that the emf is a separate force from the electric field, which is wrong.

Correct, there is no magnetic field, time varying or otherwise and thus, the electric field is conservative. This is just the way it is. The chemical forces that produce the emf are not identical to the electric field which, again, is conservative. This is not hard to see and I don't understand why you insist otherwise.