Autonomous navigation is crucial for various robotics applications in agriculture. However, many existing methods depend on RTK-GPS devices, which can be susceptible to loss of radio signal or intermittent reception of corrections from the internet. Consequently, research has increasingly focused on using RGB cameras for crop-row detection, though challenges persist when dealing with grown plants. This paper introduces a LiDAR-based navigation system that can achieve crop-agnostic over-canopy autonomous navigation in row-crop fields, even when the canopy fully blocks the inter-row spacing. Our algorithm can detect crop rows across diverse scenarios, encompassing various crop types, growth stages, illumination conditions, the presence of weeds, curved rows, and discontinuities. Without utilizing a global localization method (i.e., based on GPS), our navigation system can perform autonomous navigation in these challenging scenarios, detect the end of the crop rows, and navigate to the next crop row autonomously, providing a crop-agnostic approach to navigate an entire field. The proposed navigation system has undergone tests in various simulated and real agricultural fields, achieving an average cross-track error of 3.55 cm without human intervention. The system has been deployed on a customized UGV robot, which can be reconfigured depending on the field conditions.