IQ One Million
Because why settle for a lousy 60 standard deviations when you can have almost 70,000.
Fans of genetic engineering of the brain (like Lumo and Steve Hsu) think that human brains can be tweaked to produce IQs of 1000, whatever that might mean. Those who have studied biology are more dubious.
One component of IQ is mental processing speed, and that has got to be limited by the physiology of nerves. Fundamental neural processes take milliseconds or longer, and the basic design is not going to permit much speedup. Silicon transistors, on the other hand, are 1 million times faster.
Another component of IQ is memory, short term and long. Given enough disk space, a computer can remember everything it has ever read, heard, or seen. A number of humans have been reported to have eidetic memories, but psychologists who have attempted to study it are highly dubious, though some children have short term eidetic memories. It's physiologically improbable that a person could store all their conscious experience long term.
Guys like Steve Hsu like to make much of John von Neumann's remarkable abilities. One of the abilities attributed to him was remembering everything he read and being able to quote it back verbatim. Without a doubt von Neumann was super smart and amazingly accomplished, but did he really have that kind of recall? One first person account said something more like "he could remember perfectly a page of a book he read fifteen years ago but couldn't remember what he ate for lunch." Remembering a page of a book is far from remembering everything that he had ever read. His memory was no doubt amazing but very likely imperfect like everyone else's, only to a lesser degree. He was also said to have forgotten mathematics that he himself had developed.
Perhaps the most fundamental kind of intelligence tested by IQ tests is the ability to recognize patterns and see how to make predictions from them. This is probably an ability that does relate to the size and interconnectedness of certain brain regions - the depth of the relevant neural networks. Perhaps we can increase those by breeding humans with heads the size of science fiction aliens, but once again the very slow processing speed of the neural substrate is a limiting factor.
Humans may be able to create something with an IQ of 300 or 1000, but it's very unlikely to be human or based on biological neurons. Silicon seems far more promising.