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With the emergence of 3D Gaussian Splatting (3DGS), numerous pioneering efforts have been made to address the effective compression issue of massive 3DGS data. 3DGS offers an efficient and scalable representation of 3D scenes by utilizing learnable 3D Gaussians, but the large size of the generated data has posed significant challenges for storage and transmission. Existing methods, however, have been limited by their inability to support progressive coding, a crucial feature in streaming applications with varying bandwidth. To tackle this limitation, this paper introduce a novel approach that organizes 3DGS data into an octree structure, enabling efficient progressive coding. The proposed ProGS is a streaming-friendly codec that facilitates progressive coding for 3D Gaussian splatting, and significantly improves both compression efficiency and visual fidelity. The proposed method incorporates mutual information enhancement mechanisms to mitigate structural redundancy, leveraging the relevance between nodes in the octree hierarchy. By adapting the octree structure and dynamically adjusting the anchor nodes, ProGS ensures scalable data compression without compromising the rendering quality. ProGS achieves a remarkable 45X reduction in file storage compared to the original 3DGS format, while simultaneously improving visual performance by over 10%. This demonstrates that ProGS can provide a robust solution for real-time applications with varying network conditions.