Constant appetite, ultimately, for food (3)
I believe the answer is:
pie
'food' is the definition.
(pie is a kind of food)
'constant appetite ultimately' is the wordplay.
'constant' becomes 'pi' (mathematics: 3.14).
'ultimately' indicates one should take the final letters.
The last letter of 'appetite' is 'e'.
'pi'+'e'='PIE'
'for' acts as a link.
(Other definitions for pie that I've seen before include "Irrational things to do with circles and logs" , "bakery item" , "Anna's constituent" , "Small, former coin of India" , "Filled pastry" .)