Biography: story containing minimum of fun (4)
I believe the answer is:
life
'biography' is the definition.
(synonyms)
'story containing minimum of fun' is the wordplay.
'story' becomes 'lie' (story can mean a fabrication).
'containing' indicates putting letters inside.
'minimum of' indicates taking the first letters.
The first letter of 'fun' is 'f'.
'lie' going around 'f' is 'LIFE'.
(Other definitions for life that I've seen before include "'While there's .... there's hope'" , "Maximum stretch?" , "Vitality, vigour" , "Animation" , "and 12 Down: Biology, medicine, etc" .)