This paper introduces a cognitive architecture for a humanoid robot to engage in a proactive, mixed-initiative exploration and manipulation of its environment, where the initiative can originate from both the human and the robot. The framework, based on a biologically-grounded theory of the brain and mind, integrates a reactive interaction engine, a number of state-of-the art perceptual and motor learning algorithms, as well as planning abilities and an autobiographical memory. The architecture as a whole drives the robot behavior to solve the symbol grounding problem, acquire language capabilities, execute goal-oriented behavior, and express a verbal narrative of its own experience in the world. We validate our approach in a human-robot interaction experiment with the iCub humanoid robot, showing that the proposed cognitive architecture can be applied in real time within a realistic scenario.