A girl who grew to accept university’s stiff examination (7)
I believe the answer is:
autopsy
'stiff examination' is the definition.
(examination of a body or 'stiff')
'a girl who grew to accept university's' is the wordplay.
'girl who grew' becomes 'Topsy' ('I 'spect I growed' - Topsy in Uncle Tom's Cabin).
'to accept' means one lot of letters goes inside another.
'university' becomes 'u' (abbreviation for university).
'a'+'topsy'='atopsy'
'atopsy' placed around 'u' is 'AUTOPSY'.
(Other definitions for autopsy that I've seen before include "Examination after death, a post-mortem" , "final critical analysis" , "Post-mortem inspection" , "Examining body" , "Examination to determine cause of death" .)