Even if we did find an index of intelligence that didn't correlate with IQ/g, would we count it as such? Duckworth & Seligman discovered that in a sample of 164 schoolchildren, a composite measure of self-discipline predicted GPA significantly better than IQ, and self-discipline didn't correlate significantly with IQ. Does self-discipline now count as an independent intellectual ability? I'd lean towards saying it doesn't, but I doubt I could justify being dogmatic about that; it's surely a cognitive ability in the term's broadest sense.

Interesting thought. It turns out that Conscientiousness is actually negatively related to intelligence, while Openness is positively correlated with intelligence.

This finding is consistent with the folk notion of "crazy geniuses."

Though it's important to note that the second study was done on college students, who must have a certain level of IQ and who aren't representative of the population.

The first study notes:

According to this proposal the significant negative correlation could be observed only in groups with above average mental abilities and not in a random sample from a general population.

If we took a larger sample of the population, including lower IQ individuals, then I think we would see the negative correlation between Conscientiousness and intelligence diminish or even reverse, because I bet there are lots of people outside a college population who have both low intelligence and low Conscientiousness.

It could be that a moderate amount of Conscientiousness (well, whatever mechanisms cause Conscientiousness) is necessary for above average intelligence, but too much Conscientiousness (i.e. those mechanisms are too strong) limits intelligence.