Master enters Exeter prepared for highest degree (7)
I believe the answer is:
extreme
'highest degree' is the definition.
(thesaurus)
'master enters exeter prepared' is the wordplay.
'master' becomes 'M' (common abbreviation eg MA, Master of Art).
'enters' means one lot of letters goes inside another.
'prepared' indicates anagramming the letters.
'exeter' with letters rearranged gives 'extree'.
'm' inserted into 'extree' is 'EXTREME'.
'for' is the link.
(Other definitions for extreme that I've seen before include "Ultimate" , "Most distant" , "too much" , "over the top" , "Unrestrained" .)