Dense packing in pick-and-place systems is an important feature in many warehouse and logistics applications. Prior work in this space has largely focused on planning algorithms in simulation, but real-world packing performance is often bottlenecked by the difficulty of perceiving 3D object geometry in highly occluded, partially observed scenes. In this work, we present a fully-convolutional shape completion model, F-CON, which can be easily combined with off-the-shelf planning methods for dense packing in the real world. We also release a simulated dataset, COB-3D-v2, that can be used to train shape completion models for real-word robotics applications, and use it to demonstrate that F-CON outperforms other state-of-the-art shape completion methods. Finally, we equip a real-world pick-and-place system with F-CON, and demonstrate dense packing of complex, unseen objects in cluttered scenes. Across multiple planning methods, F-CON enables substantially better dense packing than other shape completion methods.