Enabling autonomous robots to operate robustly in challenging environments is necessary in a future with increased autonomy. For many autonomous systems, estimation and odometry remains a single point of failure, from which it can often be difficult, if not impossible, to recover. As such robust odometry solutions are of key importance. In this work a method for tightly-coupled LiDAR-Radar-Inertial fusion for odometry is proposed, enabling the mitigation of the effects of LiDAR degeneracy by leveraging a complementary perception modality while preserving the accuracy of LiDAR in well-conditioned environments. The proposed approach combines modalities in a factor graph-based windowed smoother with sensor information-specific factor formulations which enable, in the case of degeneracy, partial information to be conveyed to the graph along the non-degenerate axes. The proposed method is evaluated in real-world tests on a flying robot experiencing degraded conditions including geometric self-similarity as well as obscurant occlusion. For the benefit of the community we release the datasets presented: https://github.com/ntnu-arl/lidar_degeneracy_datasets.