Peephole optimization is an essential class of compiler optimizations that targets small, inefficient instruction sequences within programs. By replacing such suboptimal instructions with refined and more optimal sequences, these optimizations not only directly optimize code size and performance, but also enable more transformations in the subsequent optimization pipeline. Despite their importance, discovering new and effective peephole optimizations remains challenging due to the complexity and breadth of instruction sets. Prior approaches either lack scalability or have significant restrictions on the peephole optimizations that they can find. This paper introduces LPO, a novel automated framework to discover missed peephole optimizations. Our key insight is that, Large Language Models (LLMs) are effective at creative exploration but susceptible to hallucinations; conversely, formal verification techniques provide rigorous guarantees but struggle with creative discovery. By synergistically combining the strengths of LLMs and formal verifiers in a closed-loop feedback mechanism, LPO can effectively discover verified peephole optimizations that were previously missed. We comprehensively evaluated LPO within LLVM ecosystems. Our evaluation shows that LPO can successfully identify up to 22 out of 25 previously reported missed optimizations in LLVM. In contrast, the recently proposed superoptimizers for LLVM, Souper and Minotaur detected 15 and 3 of them, respectively. More importantly, within eleven months of development and intermittent testing, LPO found 62 missed peephole optimizations, of which 28 were confirmed and an additional 13 had already been fixed in LLVM. These results demonstrate LPO's strong potential to continuously uncover new optimizations as LLMs' reasoning improves.