Business errand, extremely involved (9)
I believe the answer is:
concerned
'involved' is the definition.
(similar in meaning)
'business errand extremely' is the wordplay.
'business' becomes 'concern' ('concern' can be a synonym of 'business').
'extremely' says to hollow out the word (remove centre letters).
'errand' with its centre taken out is 'ed'.
'concern'+'ed'='CONCERNED'
(Other definitions for concerned that I've seen before include "Feeling or showing worry or solicitude" , "Worried, anxious" , "Involved - worried" , "anxiety" .)