The most obvious example is the limits of reason--even pure reason. Now to Kant "pure reason" merely means not based on observation. But to me it seems the implication is clear that he was saying even pure reason in itself, not just human reason.
Red used to be thought of as in the object. Descartes noticed there are things that are not really in the object itself but depend on the subject observing. Kant noticed all characteristics depend on the subject. So what is left? The thing in itself.
Also universality and necessity as abstract ideas can not be derived by induction, no matter how many times you see them.
Kant argues against universality and necessity are not in objects, while universality and necessity are true, as in mathematics and natural science.
As Hegel puts it: "But if universality and necessity do not exist in external things, the question arises “Where are they to be found?” Kant maintains that they must be that they must rest on reason itself, and on thought as self-conscious reason; their source is the subject, “I”. This, simply expressed, is the main point in the Kantian philosophy. What makes them valid is that the object depends on the subject for its character."
Thus reason can not enter into "unconditioned reality" (things in themselves) that in no way is connected with physical objects.
To Hegel the kind of dialectical method used by Socrates is the very nature of reason in itself and allows reason to progress clearly and definitely into un-conditioned reality.
The way the Stanford Encyclopedia puts it: Kant’s mistake was that he fell short of saying that these contradictions are in the world itself. He failed to apply the insights of his discussion of the antinomies to “things in themselves” Indeed, Kant’s own argument proves that the dialectical nature of reason can be applied to things themselves. The fact that reason develops those contradictions on its own, without our heads to help it, shows that those contradictions are not just in our heads, but are objective, or in the world itself.
[In any case I should mention I learned at lot from Dr. Kelly Ross and his particular approach to Kantian philosophy. And I see Kant as being a kind of umbrella and bringing out important points.]