We study open-world multi-label text classification under extremely weak supervision (XWS), where the user only provides a brief description for classification objectives without any labels or ground-truth label space. Similar single-label XWS settings have been explored recently, however, these methods cannot be easily adapted for multi-label. We observe that (1) most documents have a dominant class covering the majority of content and (2) long-tail labels would appear in some documents as a dominant class. Therefore, we first utilize the user description to prompt a large language model (LLM) for dominant keyphrases of a subset of raw documents, and then construct a (initial) label space via clustering. We further apply a zero-shot multi-label classifier to locate the documents with small top predicted scores, so we can revisit their dominant keyphrases for more long-tail labels. We iterate this process to discover a comprehensive label space and construct a multi-label classifier as a novel method, X-MLClass. X-MLClass exhibits a remarkable increase in ground-truth label space coverage on various datasets, for example, a 40% improvement on the AAPD dataset over topic modeling and keyword extraction methods. Moreover, X-MLClass achieves the best end-to-end multi-label classification accuracy.