The first thing to come to mind is the traveling salesman problem and the host of other unsolved math problems which we suspect may be unsolvable-by-us.
There's also problems of self reference. A Turing machine may be able to solve the halting problem for pushdown automata, but it can't solve the halting problem for Turing machines. Whether or not we're as capable as Turing machines, there's a halting problem for us and we can't solve it.
I'm restricted to mathy spaces here because how else would you construct a well defined question that you cannot answer? But I see no reason why there wouldn't be other perspectives that we're incapable of accessing, it's just that in these cases the ability to construct the question is just as out of reach as the ability to answer it.
You may have heard talk about known unknowns and unknown unknowns, but there are also known unknowables and unknown unknowables, and maybe even unknowable unknowables (I go into this in greater detail here: https://github.com/MatrixManAtYrService/Righting/blob/master...).
In any case, I don't think it's ever valid to take one's inability to find examples as proof of something unless you can also prove that the search was exhaustive.
Instead of AGI, we should call it AHI: artificial human intelligence, or SHI: super human intelligence. That would be much clearer and would sidestep the generality issue.