155, 160, or maybe 171.1
The answer is different for each different IQ test.2
The most common IQ tests given to adults in the USA for the past 70-odd years are the Wechsler Adult Intelligence Scale series. The WAIS-V is currently under development, expected to be released in 2019; until then, the WAIS-IV is the standard.
On the WAIS-IV, the maximum score is 155+.
However, this is in part because the scaling process effectivly ends with "if the equation spits out a number from 155-160, record it as 155+". The WAIS-IV minus that rule would have a maximum score of 160.
Various high-IQ groups have come up with the idea of designing a test that's more challenging and, more importantly, calibrated specifically against the upper tail (e.g., against Mensa members).
Both Mensa and the Triple Nine Society (who require a higher cutoff of 146) are perfectly happy with the WAIS. But what if you wanted to start a society that required a cutoff of 171? The Prometheus and Four Sigma societies3 both wanted to restrict themselves to one-in-a-million geniuses.
Four Sigma's Kevin Langdon designed a test by basically gathering all of the hardest questions from other IQ tests, getting rid of any time limits, and calibrating it against self-selected geniuses from among the readership of Omni.
Langdom claimed to have just barely reached the goal of an accurate ceiling of 171. Prometheus went with it as well.
Years later, the Langdon test's answers were published on the internet, making it worthless. Prometheus did a comparative study of existing tests and found that a WAIS score of 155+ was actually better correlated with their membership than any of the existing high-tail tests. Which means all of those tests (not necessarily including Langdon's—but probably) are actually useless for distinguishing between 155-170 and 171+ people after all.
Now, when people write that "Einstein had an IQ above 190", what do they mean?
The meaning of IQ numbers is pretty simple: 100 is average, and each standard deviation from the average is 15 points (with the assumption that intelligence follows a normal distribution, of course). So, 190 is 6 standard deviations, about 2 in a billion. And that's basically what it means: Someone made a wild guess that Einstein was a one-in-a-billion genius, and convinced some psychology professor to do the undergrad-level math that converts "one-in-a-billion" to "over 190", and then wrote an article saying that Einstein had an IQ over 190.
The psychologist almost certainly gave the journalist a bunch of caveats about how meaningless that was—not to mention that the first four experts they tried all gave answers more like, "Well, as far as I know he never took a test, but it seems likely he'd have gotten a 155+"—but the journalist ignored all of that, because "Einstein's IQ over 190, says Harvard psychology professor" is a better story than "Journalist doesn't understand IQ but wants to write article on it anyway".
If a super-intelligent AI wanted to aggrandize itself misleadingly, it could pull the same kinds of shenanigans, but I think it would be smart enough to not bother.
If it's anything like smart humans, it's probably more self-impressed with its geek-trivia knowledge than its intelligence, so it would probably put on its best Holly voice and say, "I have an IQ of 6000, the same as 6000 PE teachers".
Or… I'm pretty sure there's a golden-age scifi story that I read as a kid (no idea which one, who it was by, or anything else, except that it was probably in one of my grandmother's old sci-fi story collections) where the world's first supercomputer says something like, "On a test calibrated for humans, I'd get a perfect score of 164. On a test calibrated for my peer group, I'd get a score of 100, because I am my only peer." That struck me as a better way to show its pompous arrogance.
1. Things are slightly different for children; some tests scale up as high as 205. But I assume the AI isn't going to want to claim to be super-smart for a 5-year-old.
2. Academics just rescale stdev=16 results as stdev=15 for comparisons between them, so let's do the same here; otherwise you're just adding confusion for no purpose. But an AI that wanted to be misleading could get away with claiming 175 on a 16-scale instead of 171 on a 15-scale.
3. Each of them being, at that point, one guy, each of whom were deciding whether to let the other guy into his clubhouse…