Completely or absolutely; to the fullest degree.
It comes from 'total', which comes from Latin 'totalis' meaning 'entire' or 'whole', plus the adverb-forming suffix '-ly'. It has been used in English since the Middle Ages.
In casual speech, 'totally' has weakened from meaning '100%' to sometimes just meaning 'yes' or 'really', as in 'Totally!' This is a classic example of how strong words in language gradually get softer over time.
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