I know that the IQ statistic is designed to give it a mean of 100 and that you'll certainly never find someone with an IQ below 1 or above 300, but that tells us very little about the variance or general shape of the distribution. So why is it that every graph of IQ scores that I've seen appears to be a truncated normal distribution? Is it some property of the test design, some property of the test subjects, or some deep theorem in statistics that I've overlooked?
IQ isn't normal, it's normalized to have mean 100 and standard deviation 15, usually via a percentile method.
The reason IQ looks roughly normal is because intelligence (however it is defined) is a complex trait. Complex traits are predicted to have a roughly normal distribution based on the central limit theorem: a sum of many individual factors (including genetic and environmental ones) will tend to be distributed normally in a population, even if the underlying factors themselves are not normal.
There is no real concrete measure "IQ": it isn't measuring a real-world physical property the way you measure mass or length. Instead, you use tests intended to get some measure of that abstract trait, and then normalize individuals based on the group statistics. Actual tests administered to measure IQ will have a minimum and maximum score: you can at worst get every question wrong, at best get every question right.