IQ Test stability longitudinal study
There are some subtleties to this question which you may or may not realize. There are (at least) two similarly sounding questions, but they are actually quite different questions:


Does an individual's general cognitive abilities change throughout life?
Does the rank-order of a population's cognitive abilities change throughout life? In other words, if we test a population at one time and then again at another time, will those who scored highly on the first test also be the ones who score highly on the second test and vice versa?


The answer to the first is trivially yes. An individual's brain is more powerful and has learned more when they are 22, then when they are 4, say. There is also reduction in cognitive abilities when people become elderly.

The second question is more interesting. For this I would refer first to the paper:

Deary et al (2000). The stability of individual differences in mental ability from childhood to old age: Follow-up of the 1932 Scottish Mental Survey. See pdf here.

They conclude that


  The correlation between Moray House Test scores at age 11 and age 77 was 0.63, which adjusted to 0.73 when corrected for attenuation of ability range within the re-tested sample. This, the longest follow-up study of psychometric intelligence reported to date, shows that mental ability differences show substantial stability from childhood to late life. 


So the rank-order of abilities within a population probably correlates >0.7 with its rank-order six and a half decades later (age 11 to age 77).

Another interesting study, which is new and I don't believe is published yet, is the following: 

Grønkjær et al. Associations between education and age-related cognitive changes in a 41-year follow-up study

In this Danish longitudinal study, they have them first tested at mean age 20, then again 41 years later at mean age 61. They report finding:


  The mean change in BPP test scores was -2.94 (SD 5.60) and a retest correlation of 0.81 between the baseline and follow-up BPP scores was observed.


These correlations are not quite in the 0.9-1.0 range, and hence we're certainly not seeing perfect stability. On the other hand, I don't think I know any quantitatively measured psychological trait which shows a greater longitudinal stability (in terms of rank-ordering) than general cognitive ability.