Increasing the density of the 3D LiDAR point cloud is appealing for many applications in robotics. However, high-density LiDAR sensors are usually costly and still limited to a level of coverage per scan (e.g., 128 channels). Meanwhile, denser point cloud scans and maps mean larger volumes to store and longer times to transmit. Existing works focus on either improving point cloud density or compressing its size. This paper aims to design a novel 3D point cloud representation that can continuously increase point cloud density while reducing its storage and transmitting size. The pipeline of the proposed Continuous, Ultra-compact Representation of LiDAR (CURL) includes four main steps: meshing, upsampling, encoding, and continuous reconstruction. It is capable of transforming a 3D LiDAR scan or map into a compact spherical harmonics representation which can be used or transmitted in low latency to continuously reconstruct a much denser 3D point cloud. Extensive experiments on four public datasets, covering college gardens, city streets, and indoor rooms, demonstrate that much denser 3D point clouds can be accurately reconstructed using the proposed CURL representation while achieving up to 80% storage space-saving. We open-source the CURL codes for the community.