Identifying the small subset of source code that repeatedly attracts bugs is critical for reducing long-term maintenance effort. We define ExtremelyBuggy methods as those involved in more than one bug fix and present the first large-scale study of their prevalence, characteristics, and predictability. Using a dataset of over 1.25 million methods from 98 open-source Java projects, we find that ExtremelyBuggy methods constitute only a tiny fraction of all methods, yet frequently account for a disproportionately large share of bugs. At their inception, these methods are significantly larger, more complex, less readable, and less maintainable than both singly-buggy and non-buggy methods. However, despite these measurable differences, a comprehensive evaluation of five machine learning models shows that early prediction of ExtremelyBuggy methods remains highly unreliable due to data imbalance, project heterogeneity, and the fact that many bugs emerge through subsequent evolution rather than initial implementation. To complement these quantitative findings, we conduct a thematic analysis of 265 ExtremelyBuggy methods, revealing recurring visual issues (e.g., confusing control flow, poor readability), contextual roles (e.g., core logic, data transformation, external resource handling), and common defect patterns (e.g., faulty conditionals, fragile error handling, misuse of variables). These results highlight the need for richer, evolution-aware representations of code and provide actionable insights for practitioners seeking to prioritize high-risk methods early in the development lifecycle.

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