Despite being a common figure of speech, hyperbole is under-researched with only a few studies addressing its identification task. In this paper, we introduce a new task of hyperbole generation to transfer a literal sentence into its hyperbolic paraphrase. To tackle the lack of available hyperbolic sentences, we construct HYPO-XL, the first large-scale hyperbole corpus containing 17,862 hyperbolic sentences in a non-trivial way. Based on our corpus, we propose an unsupervised method for hyperbole generation with no need for parallel literal-hyperbole pairs. During training, we fine-tune BART to infill masked hyperbolic spans of sentences from HYPO-XL. During inference, we mask part of an input literal sentence and over-generate multiple possible hyperbol