Mixture-of-Experts (MoE), while offering significant advantages as a Large Language Model (LLM) architecture, faces substantial challenges when deployed on low-cost edge devices with tight memory constraints. Expert offloading mitigates this issue by storing expert parameters in CPU memory and caching a subset of popular experts in GPU memory. Although this approach improves GPU memory utilization by caching only the likely-used experts, the GPU memory reserved for expert caching is underutilized compared with dense LLMs. This paper presents OD-MoE, a distributed MoE inference framework that obviates the need for expert caches via fully on-demand expert loading. OD-MoE is built upon two key mechanisms: 1) parallelizing expert loading and expert computation across distributed edge nodes, and 2) an ultra-accurate emulative predictor that forecasts expert activations multiple layers ahead while expert computation is ongoing. With these innovations, OD-MoE dynamically loads each target expert to one of the distributed nodes just-in-time before its activation and promptly evicts it afterward, freeing GPU memory for subsequent experts. We comprehensively benchmark OD-MoE against state-of-the-art MoE offloading systems on a ten-node testbed. Experimental results show that: 1) OD-MoE achieves 99.94% expert activation prediction accuracy, substantially surpassing all existing methods; and 2) OD-MoE delivers approximately 75% of the decoding speed of a fully GPU-cached MoE deployment while using only 1/3 of the GPU memory. More importantly, by eliminating the need for expert caches, OD-MoE enables MoE inference on edge nodes with less-than-1GB GPU memory, paving the way for practical MoE deployment of low-cost IoT devices at the edge in the LLM era.