Characterized by or prone to babbling; talkative, chatty, or producing streams of incoherent or trivial talk.
From 'babble' plus the suffix -y, meaning 'characterized by.' The -y suffix is one of English's most common adjectival endings, creating words like 'happy,' 'sleepy,' and 'busy' from nouns and verbs.
The -y suffix is so productive in English that almost any noun can become an adjective—'water' becomes 'watery,' 'sleep' becomes 'sleepy'—which is why 'babbly' feels immediately understandable even if you've never heard it before.
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