Language processing involves the ability to master supra-regular grammars, that go beyond the level of complexity of regular grammars. This ability has been hypothesized to be a uniquely human capacity. Our study probed baboons' capacity to learn two supra-regular grammars of different levels of complexity: a context-free grammar generating sequences following a mirror structure (e.g., AB | BA, ABC | CBA) and a context-sensitive grammar generating sequences following a repeat structure (e.g., AB | AB, ABC | ABC), the latter requiring greater computational power to be processed. Fourteen baboons were tested in a prediction task, requiring them to track a moving target on a touchscreen. In distinct experiments, sequences of target locations followed one of the above two grammars, with rare violations. Baboons showed slower response times when violations occurred in mirror sequences, but did not react to violations in repeat sequences, suggesting that they learned the context-free (mirror) but not the context-sensitive (repeat) grammar. By contrast, humans tested with the same task learned both grammars. These data suggest a difference in sensitivity in baboons between a context-free and a context-sensitive grammar.