Compositionality is the principle that a system should be designed by composing together smaller subsystems, and reasoning about the system should be done recursively on its structure.
Clearly interfaces are a crucial aspect of compositionality, and I suspect that interfaces are in fact synonymous with compositionality. That is, compositionality is not just the ability to compose objects, but the ability to work with an object after intentionally forgetting how it was built. The part that is remembered is the ‘interface’, which may be a type, or a contract, or some other high-level description. The crucial property of interfaces is that their complexity stays roughly constant as systems get larger.
In software, for example, an interface can be used without knowing whether it represents an atomic object, or a module containing millions of lines of code whose implementation is distributed over a large physical network.
For examples of non-compositional systems, we look to nature. Generally speaking, the reductionist methodology of science has difficulty with biology, where an understanding of one scale often does not translate to an understanding on a larger scale.
For example, the behaviour of neurons is well-understood, but groups of neurons are not.
More generally, I claim that the opposite of compositionality is emergent effects. The common definition of emergence is a system being ‘more than the sum of its parts’, and so it is easy to see that such a system cannot be understood only in terms of its parts, i.e. it is not compositional. Moreover I claim that non-compositionality is a barrier to scientific understanding, because it breaks the reductionist methodology of always dividing a system into smaller components and translating explanations into lower levels.