This article presents a risk analysis of large language models (LLMs), a type of "generative" artificial intelligence (AI) system that produces text, commonly in response to textual inputs from human users. The article is specifically focused on the risk of LLMs causing an extreme catastrophe in which they do something akin to taking over the world and killing everyone. The possibility of LLM takeover catastrophe has been a major point of public discussion since the recent release of remarkably capable LLMs such as ChatGPT and GPT-4. This arguably marks the first time when actual AI systems (and not hypothetical future systems) have sparked concern about takeover catastrophe. The article's analysis compares (A) characteristics of AI systems that may be needed for takeover, as identified in prior theoretical literature on AI takeover risk, with (B) characteristics observed in current LLMs. This comparison reveals that the capabilities of current LLMs appear to fall well short of what may be needed for takeover catastrophe. Future LLMs may be similarly incapable due to fundamental limitations of deep learning algorithms. However, divided expert opinion on deep learning and surprise capabilities found in current LLMs suggests some risk of takeover catastrophe from future LLMs. LLM governance should monitor for changes in takeover characteristics and be prepared to proceed more aggressively if warning signs emerge. Unless and until such signs emerge, more aggressive governance measures may be unwarranted.