Anecdote beginning to shock Conservative (5)
I believe the answer is:
story
'anecdote' is the definition.
(an anecdote is a funny story)
'beginning to shock conservative' is the wordplay.
'beginning to' suggests taking the first letters.
'conservative' becomes 'tory' (informal term for a Conservative Party member).
The first letter of 'shock' is 's'.
's'+'tory'='STORY'
(Other definitions for story that I've seen before include "Account, tale" , "Rumour" , "Tale - lie" , "See 10 and 19" , "Narration" .)