What is the known cheapest chemical solution to extract CO2 from air on industrial scale? As far as I know, the cheapest non-chemical solution is: grow grass, and burn it. The resulting $\ce{CO2}$ is from the air. But I am interested on chemical industrial solutions now.

As far as I know, the main difficulty of the process is that the ratio of the $\ce{CO2}$ in the air is very low. Thus, extracting it means a large entropy difference.

My layman's intuition is that maybe some substance exists - probably an organic compound - which binds especially strongly to $\ce{CO2}$ and it is easily regenerated. One would pump air through it, and then regenerate it to get the $\ce{CO2}$ back. If such a compound exists, what is it?

The process should be powered by electrical energy only. Thus, solutions such as "buy gas, burn it, and you get much cheaper $\ce{CO2}$" are not acceptable.

On "cheapest" I understand, what is cheapest in industrial size, not in a laboratory? (for example, in a large chemical factory)
Your intuition is correct, and such compounds are plenty. The cheapest are probably $\ce{NaOH}$, $\ce{Ca(OH)_2}$ and the like, but these are not easily regenerable. Then there are organic amines, like monoethanolamine, which can do the same in a reversible manner. Indeed, they are used in industry.