Robust initialization is crucial for online systems. In the letter, a high-frequency and resilient initialization framework is designed for LiDAR-inertial systems, leveraging both inertial sensors and Doppler LiDAR. The innovative FMCW Doppler LiDAR opens up a novel avenue for robotic sensing by capturing not only point range but also Doppler velocity via the intrinsic Doppler effect. By fusing point-wise Doppler velocity with inertial measurements under non-inertial kinematics, the proposed framework, Free-Init, eliminates reliance on motion undistortion of LiDAR scans, excitation motions, and map correspondences during the initialization phase. Free-Init is also plug-and-play compatible with typical LiDAR-inertial systems and is versatile to handle a wide range of initial motions when the system starts, including stationary, dynamic, and even violent motions. The embedded Doppler-inertial velocimeter ensures fast convergence and high-frequency performance, delivering outputs exceeding 10 kHz. Comprehensive experiments on diverse platforms and across myriad motion scenes validate the framework's effectiveness. The results demonstrate the superior performance of Free-Init, highlighting the necessity of fast, resilient, and dynamic initialization for online systems.