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Many clinical and research studies of the human brain require an accurate structural MRI segmentation. While traditional atlas-based methods can be applied to volumes from any acquisition site, recent deep learning algorithms ensure very high accuracy only when tested on data from the same sites exploited in training (i.e., internal data). The performance degradation experienced on external data (i.e., unseen volumes from unseen sites) is due to the inter-site variabilities in intensity distributions induced by different MR scanner models, acquisition parameters, and unique artefacts. To mitigate this site-dependency, often referred to as the scanner effect, we propose LOD-Brain, a 3D convolutional neural network with progressive levels-of-detail (LOD) able to segment brain data from any site. Coarser network levels are responsible to learn a robust anatomical prior useful for identifying brain structures and their locations, while finer levels refine the model to handle site-specific intensity distributions and anatomical variations. We ensure robustness across sites by training the model on an unprecedented rich dataset aggregating data from open repositories: almost 27,000 T1w volumes from around 160 acquisition sites, at 1.5 - 3T, from a population spanning from 8 to 90 years old. Extensive tests demonstrate that LOD-Brain produces state-of-the-art results, with no significant difference in performance between internal and external sites, and robust to challenging anatomical variations. Its portability opens the way for large scale application across different healthcare institutions, patient populations, and imaging technology manufacturers. Code, model, and demo are available at the project website.