Continual learning describes a setting where machine learning models learn novel concepts from continuously shifting training data, while simultaneously avoiding degradation of knowledge on previously seen classes (a phenomenon known as the catastrophic forgetting problem) which may disappear from the training data for extended periods of time. Current approaches for continual learning of a single expanding task (aka class-incremental continual learning) require extensive rehearsal of previously seen data to avoid this degradation of knowledge. Unfortunately, rehearsal comes at a sharp cost to memory and computation, and it may also violate data-privacy. Instead, we explore combining knowledge distillation and parameter regularization in new ways to achieve strong continual learning performance without rehearsal. Specifically, we take a deep dive into common continual learning techniques: prediction distillation, feature distillation, L2 parameter regularization, and EWC parameter regularization. We first disprove the common assumption that parameter regularization techniques fail for rehearsal-free continual learning of a single, expanding task. Next, we explore how to leverage knowledge from a pre-trained model in rehearsal-free continual learning and find that vanilla L2 parameter regularization outperforms EWC parameter regularization and feature distillation. We then highlight the impact of the rehearsal-free continual learning settings with a classifier expansion benchmark, showing that a strategy based on our findings combined with a positive/negative label balancing heuristic can close the performance gap between the upper bound and the existing strategies by up to roughly 50%. Finally, we show that a simple method consisting of pre-training, L2 regularization, and prediction distillation can even outperform rehearsal-based methods on the common CIFAR-100 benchmark.