Royal house's beginning to encompass university (6)
I believe the answer is:
stuart
'royal house's' is the definition.
(royal house of Scotland and England)
'beginning to encompass university' is the wordplay.
'beginning' becomes 'start' (synonyms).
'to encompass' means one lot of letters goes inside another (I've seen this in other clues).
'university' becomes 'u' (abbreviation for university).
'start' placed around 'u' is 'STUART'.
(Other definitions for stuart that I've seen before include "Member of royal family" , "Scottish royal family" , "Royal family up to seventeen fourteen" , "Name of royal family" , "C17 royal house" .)