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Notes -
... okay
As you said, 250 IQ is 10 SD. 5 SD and 10 SD are very different. If you had said 175 IQ, I would not have said that was categorically impossible. While I still do think that IQ is going to have construct issues, if you just assume a normal distribution there will be people who have + 5 SD out of 9 billion people.
The thing, though, is that intelligence isn't like muscle mass. In two ways. One (pretty confident), it's not just a thing you can measure. We genuinely do not know what the capabilities of '250 IQ' would refer to. No such people exist, not even close. IQ is defined by rank ordering an existing distribution and mapping it to a normal. If you carried out your method, and the 250 IQ person wasn't actually practically smarter than the 160 IQ person, you could still say - well, he's 250 IQ, because of the distribution of test scores we inferred the way we did the selection! And there wouldn't actually be anything wrong with that, other than the person not actually being usefully smarter.
Two, (much less confident, plausible but not more likely than not imo that the scientific consensus disagrees with this) intelligence isn't something where there's an obvious disadvantage to more of it. For muscle mass, past a certain point there's some fitness advantage from being stronger but it's more than compensated for by things like energy / diet requirements. So in an artificial environment with infinite food, it's really easy for natural selection to just modify whatever regulates how much muscles grow and grow more and have them get massive. I think intelligence is just hard, though. It's just a very complicated thing, and there's no simple way to have more of it if some other tradeoff is fixed. More intelligence mostly isn't a matter of increasing the number of neurons, plenty of people with the same head size as von Neumann just weren't von Neumann. And at von Neumann's scale non-additive effects probably play a significant role in getting you from 'very smart' to 'top 100', and natural selection just won't work as well on those.
We can get freakish results via artificial selection, but yes we hit limits, eg greyhounds are only like 30% faster than wolves.
But look at corn vs teosinte, or milk yields per cow doubling in the last 50 years... or von Neumann in 1000 years from a hybrid of Middle easterners and Europeans...
I think we could make IQ tests that went further than the current highest scores ie reverse digit span tests, reaction times etc. If you can reliably get the same answers as someone with 160 IQ but faster... you're smarter than them
But yeah the real proof would be accomplishment, IQ is just the best measure we have for intelligence and the goal would be better output eg important original research etc
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