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LiDAR depth-only completion is a challenging task to estimate dense depth maps only from sparse measurement points obtained by LiDAR. Even though the depth-only methods have been widely developed, there is still a significant performance gap with the RGB-guided methods that utilize extra color images. We find that existing depth-only methods can obtain satisfactory results in the areas where the measurement points are almost accurate and evenly distributed (denoted as normal areas), while the performance is limited in the areas where the foreground and background points are overlapped due to occlusion (denoted as overlap areas) and the areas where there are no measurement points around (denoted as blank areas) since the methods have no reliable input information in these areas. Building upon these observations, we propose an effective Coupled U-Net (CU-Net) architecture for depth-only completion. Instead of directly using a large network for regression, we employ the local U-Net to estimate accurate values in the normal areas and provide the global U-Net with reliable initial values in the overlap and blank areas. The depth maps predicted by the two coupled U-Nets are fused by learned confidence maps to obtain final results. In addition, we propose a confidence-based outlier removal module, which removes outliers using simple judgment conditions. Our proposed method boosts the final results with fewer parameters and achieves state-of-the-art results on the KITTI benchmark. Moreover, it owns a powerful generalization ability under various depth densities, varying lighting, and weather conditions.