LiDAR scene completion is a key component of 3D perception in autonomous driving, where the scene must be completed in real time to be usable in downstream tasks. Existing approaches typically follow an initialize-and-refine paradigm, in which a coarse initialization of the scene is first constructed, then refined into complete 3D geometry. Generative models are slower because they iteratively refine random Gaussian noise into the scene, while non-generative methods perturb the partial scene with a fixed noise scale, which limits coverage of large gaps and occluded regions and requires manual recalibration for each new sensor configuration. We present RapidLiDAR, a LiDAR scene completion method that treats the initialization itself as a learned, data-driven component. We propose an adaptive initialization module that predicts a spatially varying displacement for each partial input point, expanding the partial observations into a coarse scene initialization adapted to the local geometry, without requiring manual noise tuning. To refine this coarse initialization into a complete and coherent scene, we additionally propose a multi-scale reconstruction module that further refines point positions by querying multi-scale 3D voxel and 2D BEV feature maps constructed from the input scan. By replacing point-neighborhood operators such as farthest point sampling and k-nearest neighbor search with voxel- and BEV-based feature extraction, our architecture is faster and can handle different input resolutions by design. Experiments on SemanticKITTI and KITTI-360 show that our method achieves completion performance on par with the state of the art while completing a full scene in 0.1 seconds, which is 2.3 times faster than the fastest prior method. This matches the 10 Hz acquisition rate of typical automotive LiDAR sensors, taking a step toward real-time LiDAR scene completion.