Monitoring snow depth is important for applications such as hydrology, energy planning, ecology, and safety evaluation for outdoor winter activities. Most methods able to estimate snow depth for large regions can only do so in a spatial resolution of up to 1 km ground sampling distance (GSD). This limits their usage in high alpine areas, where this resolution fails to capture local snow distribution patterns caused by the pronounced topographical features. In this work we use a recurrent convolutional neural network to estimate snow depth at high spatial resolution (10 m GSD), weekly, and at large scale based on satellite data sources and elevation maps, without the need for measurement stations on the ground. The proposed method achieves unprecedented results for large-scale, high-resolution snow depth mapping. The resulting maps are evaluated over a period of three years against high-fidelity snow depth maps obtained with airborne photogrammetry. Finally, we also produce well-calibrated uncertainty estimates for every individual snow depth estimate via a probabilistic regression framework.