The peak account written by the first person (4)
I believe the answer is:
acme
'the peak' is the definition.
('acme' can be a synonym of 'peak')
'account written by the first person' is the wordplay.
'account' becomes 'ac' (abbreviation).
'written by' means one lot of letters go next to another.
'the first person' becomes 'me' (first-person pronoun).
'ac'+'me'='ACME'
(Other definitions for acme that I've seen before include "Culmination" , "Summit of achievement" , "Top or highest point" , "Point of maximum development" , "high peak" .)