The present participle of 'bib,' meaning to drink or to put on/wear a bib.
From verb 'bib' + present participle suffix '-ing'. 'Bib' comes from Latin 'bibere' (to drink).
In older texts, 'bibbing' describes social drinking activities the way we might now say 'having a drink' or 'going out for drinks'—one word replacing a phrase.
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