Motivated by the need for accelerating text entry in augmentative and alternative communication (AAC) for people with severe motor impairments, we propose a paradigm in which phrases are abbreviated aggressively as primarily word-initial letters. Our approach is to expand the abbreviations into full-phrase options by leveraging conversation context with the power of pretrained large language models (LLMs). Through zero-shot, few-shot, and fine-tuning experiments on four public conversation datasets, we show that for replies to the initial turn of a dialog, an LLM with 64B parameters is able to exactly expand over 70% of phrases with abbreviation length up to 10, leading to an effective keystroke saving rate of up to about 77% on these exact expansions. Including a small amount of context in the form of a single conversation turn more than doubles abbreviation expansion accuracies compared to having no context, an effect that is more pronounced for longer phrases. Additionally, the robustness of models against typo noise can be enhanced through fine-tuning on noisy data.