Human culture relies on collective innovation: our ability to continuously explore how existing elements in our environment can be combined to create new ones. Language is hypothesized to play a key role in human culture, driving individual cognitive capacities and shaping communication. Yet the majority of models of collective innovation assign no cognitive capacities or language abilities to agents. Here, we contribute a computational study of collective innovation where agents are Large Language Models (LLMs) that play Little Alchemy 2, a creative video game originally developed for humans that, as we argue, captures useful aspects of innovation landscapes not present in previous test-beds. We, first, study an LLM in isolation and discover that it exhibits both useful skills and crucial limitations. We, then, study groups of LLMs that share information related to their behaviour and focus on the effect of social connectivity on collective performance. In agreement with previous human and computational studies, we observe that groups with dynamic connectivity out-compete fully-connected groups. Our work reveals opportunities and challenges for future studies of collective innovation that are becoming increasingly relevant as Generative Artificial Intelligence algorithms and humans innovate alongside each other.