Frenchman in Exeter could become intense (7)
I believe the answer is:
extreme
'intense' is the definition.
(similar in meaning)
'frenchman in exeter could become' is the wordplay.
'frenchman' becomes 'm' (abbreviation for Monsieur - like 'Mr').
'in' is an insertion indicator.
'could become' indicates an anagram.
'exeter' anagrammed gives 'extree'.
'm' going within 'extree' is 'EXTREME'.
(Other definitions for extreme that I've seen before include "Of the greatest intensity" , "lacking moderation" , "Very serious" , "Utmost; excessive" , "Longer than usual" .)