Grew up and dreamt about accepting university (7)
I believe the answer is:
matured
'grew up' is the definition.
(to grow up is to mature)
'dreamt about accepting university' is the wordplay.
'about' indicates an anagram.
'accepting' indicates putting letters inside.
'university' becomes 'u' (abbreviation for university).
'dreamt' anagrammed gives 'matred'.
'matred' enclosing 'u' is 'MATURED'.
'and' is the link.
(Other definitions for matured that I've seen before include "Ripened" , "Developed; grew up" , "Became fully ripe" , "Developed fully" , "Mellowed" .)