How much information do NLP tasks really need from a transformer's attention mechanism at application-time (inference)? From recent work, we know that there is sparsity in transformers and that the floating-points within its computation can be discretized to fewer values with minimal loss to task accuracies. However, this requires retraining or even creating entirely new models, both of which can be expensive and carbon-emitting. Focused on optimizations that do not require training, we systematically study the full range of typical attention values necessary. This informs the design of an inference-time quantization technique using both pruning and log-scaled mapping which produces only a few (e.g. $2^3$) unique values. Over the tasks of question answering and sentiment analysis, we find nearly 80% of attention values can be pruned to zeros with minimal ($< 1.0\%$) relative loss in accuracy. We use this pruning technique in conjunction with quantizing the attention values to only a 3-bit format, without retraining, resulting in only a 0.8% accuracy reduction on question answering with fine-tuned RoBERTa.