Initially acceptable, telling inappropriate jokes finally grows old (4)
I believe the answer is:
ages
'grows old' is the definition.
('age' can be a synonym of 'grow old')
'initially acceptable telling inappropriate jokes finally' is the wordplay.
'initially' indicates taking the first letters.
'finally' indicates one should take the final letters.
The first letter of 'acceptable' is 'a'.
The last letters of 'telling inappropriate jokes' are 'ges'.
'a'+'ges'='AGES'
(Other definitions for ages that I've seen before include "Aeons" , "Gets older" , "Rock of - - (hymn)" , "(What seems) a very long time" , "Indefinitely long time" .)