I keep seeing claims about people with IQs of 200 or even higher and I want to know if those numbers are even meaningful. I understand that IQ is normed to a population distribution but at some point the scale has to run out of reliable data right? What is the actual highest score a legitimate IQ test can produce and how seriously should we take claims about scores above 160 or 170?
This is where the statistics get really important and most people miss it. IQ scores are derived from population norming samples and the reliability of any score depends on having enough people at that level in the norming data to produce a stable estimate. At the extreme right tail, above roughly 145 or so, norming samples get very thin and the confidence intervals around any score become enormous. A score of 160 on a standard assessment might have a true score confidence interval spanning 20 or more points in either direction, which makes the precise number essentially meaningless. Most legitimate assessments cap their reported scores around 160 not because human ability cannot exceed that level but because the measurement reliability above that threshold is too poor to justify reporting a more extreme number. Claims of 200 plus are almost always derived from ratio IQ calculations, the old mental age divided by chronological age formula, which breaks down completely and produces absurdly inflated numbers that cannot be compared to modern deviation IQ scores 
The higher the claimed score the more skeptical you should be 
Legitimate psychometricians get more cautious and add more caveats as scores get more extreme. Anyone confidently claiming a precise IQ above 160 is either using a bad test or misunderstanding what their score actually means 