Imitation Learning (IL) is one of the most widely used methods in machine learning. Yet, many works find it is often unable to fully recover the underlying expert behavior, even in constrained environments like single-agent games. However, none of these works deeply investigate the role of scaling up the model and data size. Inspired by recent work in Natural Language Processing (NLP) where "scaling up" has resulted in increasingly more capable LLMs, we investigate whether carefully scaling up model and data size can bring similar improvements in the imitation learning setting for single-agent games. We first demonstrate our findings on a variety of Atari games, and thereafter focus on the extremely challenging game of NetHack. In all games, we find that IL loss and mean return scale smoothly with the compute budget (FLOPs) and are strongly correlated, resulting in power laws for training compute-optimal IL agents. Finally, we forecast and train several NetHack agents with IL and find they outperform prior state-of-the-art by 1.5x in all settings. Our work both demonstrates the scaling behavior of imitation learning in a variety of single-agent games, as well as the viability of scaling up current approaches for increasingly capable agents in NetHack, a game that remains elusively hard for current AI systems.