Digital cameras consume ~0.1 microjoule per pixel to capture and encode video, resulting in a power usage of ~20W for a 4K sensor operating at 30 fps. Imagining gigapixel cameras operating at 100-1000 fps, the current processing model is unsustainable. To address this, physical layer compressive measurement has been proposed to reduce power consumption per pixel by 10-100X. Video Snapshot Compressive Imaging (SCI) introduces high frequency modulation in the optical sensor layer to increase effective frame rate. A commonly used sampling strategy of video SCI is Random Sampling (RS) where each mask element value is randomly set to be 0 or 1. Similarly, image inpainting (I2P) has demonstrated that images can be recovered from a fraction of the image pixels. Inspired by I2P, we propose Ultra-Sparse Sampling (USS) regime, where at each spatial location, only one sub-frame is set to 1 and all others are set to 0. We then build a Digital Micro-mirror Device (DMD) encoding system to verify the effectiveness of our USS strategy. Ideally, we can decompose the USS measurement into sub-measurements for which we can utilize I2P algorithms to recover high-speed frames. However, due to the mismatch between the DMD and CCD, the USS measurement cannot be perfectly decomposed. To this end, we propose BSTFormer, a sparse TransFormer that utilizes local Block attention, global Sparse attention, and global Temporal attention to exploit the sparsity of the USS measurement. Extensive results on both simulated and real-world data show that our method significantly outperforms all previous state-of-the-art algorithms. Additionally, an essential advantage of the USS strategy is its higher dynamic range than that of the RS strategy. Finally, from the application perspective, the USS strategy is a good choice to implement a complete video SCI system on chip due to its fixed exposure time.