Fairytale beast to outstay welcome? (6)

I believe the answer is:

dragon

I'm a little stuck... Click here to teach me more about this clue!

'fairytale beast' is the definition.
'dragon' can be an answer for 'beast' (I have seen 'Mythical beast ' mean 'dragon' so perhaps 'beast' could also mean 'dragon'). I am not sure about the 'fairytale' bit.

I can't explain the remainder of the clue.

Can you help me to learn more?

(Other definitions for dragon that I've seen before include "particularly intimidating woman" , "St George's slew one" , "Fire-breathing, winged mythical monster" , "Flying monster of Welsh legend" , "Teutonic mythological creature" .)

I've seen this clue in The Times.

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