Continual Learning (CL) aims to sequentially train models on streams of incoming data that vary in distribution by preserving previous knowledge while adapting to new data. Current CL literature focuses on restricted access to previously seen data, while imposing no constraints on the computational budget for training. This is unreasonable for applications in-the-wild, where systems are primarily constrained by computational and time budgets, not storage. We revisit this problem with a large-scale benchmark and analyze the performance of traditional CL approaches in a compute-constrained setting, where effective memory samples used in training can be implicitly restricted as a consequence of limited computation. We conduct experiments evaluating various CL sampling strategies, distillation losses, and partial fine-tuning on two large-scale datasets, namely ImageNet2K and Continual Google Landmarks V2 in data incremental, class incremental, and time incremental settings. Through extensive experiments amounting to a total of over 1500 GPU-hours, we find that, under compute-constrained setting, traditional CL approaches, with no exception, fail to outperform a simple minimal baseline that samples uniformly from memory. Our conclusions are consistent in a different number of stream time steps, e.g., 20 to 200, and under several computational budgets. This suggests that most existing CL methods are particularly too computationally expensive for realistic budgeted deployment.