AI advancements have been significantly driven by a combination of foundation models and curiosity-driven learning aimed at increasing capability and adaptability. A growing area of interest within this field is Open-Endedness - the ability of AI systems to continuously and autonomously generate novel and diverse artifacts or solutions. This has become relevant for accelerating scientific discovery and enabling continual adaptation in AI agents. This position paper argues that the inherently dynamic and self-propagating nature of Open-Ended AI introduces significant, underexplored risks, including challenges in maintaining alignment, predictability, and control. This paper systematically examines these challenges, proposes mitigation strategies, and calls for action for different stakeholders to support the safe, responsible and successful development of Open-Ended AI.