Additive manufacturing (AM) increasingly relies on artificial intelligence (AI) to enable real-time monitoring, adaptive control, and process optimization, yet the deployment of advanced models remains constrained by latency, resource limitations, data security, and domain variability in industrial environments. While edge computing supports real-time responsiveness and cloud computing enables large-scale learning, their isolated use leads to fragmented intelligence and limits system-level performance. This paper proposes an Agentic Edge-Cloud Continuum (AECC) framework that integrates foundation models and hierarchical AI agents across physical, edge, fog, and cloud layers to enable coordinated data processing, model adaptation, and decision-making. In this framework, foundation models are pretrained at the cloud using distributed multi-factory data through approaches such as federated learning, adapted into domain-specific, high-fidelity models at the fog layer, and further compressed for real-time deployment at the edge alongside low-fidelity simulations for physics-based validation. The coordinated data, model, and decision flows improve efficiency by aligning computation with system constraints, enhance reliability under domain variability through continuous adaptation, and support data security through distributed data management. By addressing the fragmentation of intelligence in current systems, the proposed framework provides a structured pathway toward scalable and deployable AI-driven AM. As a conceptual foundation, it also highlights the need for further research on communication protocols, foundation model development, and adaptive model updating strategies to realize its full potential.