This paper explores speculative speech recognition (SSR), where we empower conventional automatic speech recognition (ASR) with speculation capabilities, allowing the recognizer to run ahead of audio. We introduce a metric for measuring SSR performance and we propose a model which does SSR by combining a RNN-Transducer-based ASR system with an audio-prefixed language model (LM). The ASR system transcribes ongoing audio and feeds the resulting transcripts, along with an audio-dependent prefix, to the LM, which speculates likely completions for the transcriptions. We experiment with a variety of ASR datasets on which show the efficacy our method and the feasibility of SSR as a method of reducing ASR latency.