For obvious ethical reasons, we have never perfectly conducted this experiment, but Nicaraguan Sign Language comes close. The government created a school for deaf children who did not have a common language. Many of the children did not know any language prior to attending. Although, they did have "home sign" systems, which are basic signing systems spontaneously developed between deaf children and their speaking care takers.
We can also look at creole languages. When distinct linguistic groups come into contact without a common language they will, in some cases, develop "pidgin" languages to communicate. These languages are created by the adults who are already fluent speakers of their native language. What is interesting is what happens with children born into such communities. They grow up hearing the pidgin language, but they do not learn the pidgin. Instead, the language they learn is a creole. It is based on the pidgin, but more complete and consistent with the way human language works in general. In effect, like all children, they are relying on their innate knowledge of language. The pidgin fills in the language specific parameters, but when the pidgin contradicts an aspect of innate language the children (for the most part), just don't aquire that portion of the language.
The common thread in both cases is (as you say) children: in both creoles and NSL, the early versions of the language are somewhat primitive (being developed by older children and adults), but when children are exposed to it from an early age, the languages acquire much more complexity and sophistication, becoming every bit as rich as "regular" languages. It fits well with the hypothesis that the early years of life are crucial for language development, and that there is indeed something innate in most humans for deep structural understanding of language. It's not just statistical pattern recognition.
We can also look at creole languages. When distinct linguistic groups come into contact without a common language they will, in some cases, develop "pidgin" languages to communicate. These languages are created by the adults who are already fluent speakers of their native language. What is interesting is what happens with children born into such communities. They grow up hearing the pidgin language, but they do not learn the pidgin. Instead, the language they learn is a creole. It is based on the pidgin, but more complete and consistent with the way human language works in general. In effect, like all children, they are relying on their innate knowledge of language. The pidgin fills in the language specific parameters, but when the pidgin contradicts an aspect of innate language the children (for the most part), just don't aquire that portion of the language.