Natural-language dialog is key for intuitive humanrobot interaction. It can be used not only to express humans' intents, but also to communicate instructions for improvement if a robot does not understand a command correctly. Of great importance is to endow robots with the ability to learn from such interaction experience in an incremental way to allow them to improve their behaviors or avoid mistakes in the future. In this paper, we propose a system to achieve incremental learning of complex behavior from natural interaction, and demonstrate its implementation on a humanoid robot. Building on recent advances, we present a system that deploys Large Language Models (LLMs) for high-level orchestration of the robot's behavior, based on the idea of enabling the LLM to generate Python statements in an interactive console to invoke both robot perception and action. The interaction loop is closed by feeding back human instructions, environment observations, and execution results to the LLM, thus informing the generation of the next statement. Specifically, we introduce incremental prompt learning, which enables the system to interactively learn from its mistakes. For that purpose, the LLM can call another LLM responsible for code-level improvements of the current interaction based on human feedback. The improved interaction is then saved in the robot's memory, and thus retrieved on similar requests. We integrate the system in the robot cognitive architecture of the humanoid robot ARMAR-6 and evaluate our methods both quantitatively (in simulation) and qualitatively (in simulation and real-world) by demonstrating generalized incrementally-learned knowledge.