Just how Intelligent is Kirk?

[Elder Knight]

I suspect that some kind of gene pool stagnation set in, in the future. By random chance, the Enterprise got the only seven people (more or less) who could still reason independently.

 

[BillJ]

In Court Martial it was said only three people had the ability to reprogram the ships computer... Ben Finney, Spock and Jim Kirk.

So I'd say he's very intelligent.