Daily grind almost took off bit (3,4)
I believe the answer is:
rat race
'daily grind' is the definition.
(I've seen this in another clue)
'almost took off bit' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'took off' becomes 'ran' (I've seen this in other clues).
'bit' becomes 'trace' (I've seen this in another clue).
'ran' with its last letter taken off is 'ra'.
'ra'+'trace'='RAT RACE'
(Other definitions for rat race that I've seen before include "Working life's struggle" , "keen contest" , "Continual routine of highly competitive activity" , "Competitive struggle for wealth or power" , "Grim struggle" .)