Postgres supports native nested structures. Well, as long as your structures are not circular.
JSON everywhere is really a bad choice for most data structures. It's like having no structure at all.
Well, I'm arguing that flexible schema is better than no schema. And in practice schemaless is only about schema defined outside of database. It's a solution to the wrong problem.
Also, Posgres already has hstore (already mentioned) and XML columns, which are schemaless and "unstructured".