Greek character, quiet, with space to grow rapidly (8)
I believe the answer is:
mushroom
'grow rapidly' is the definition.
(I've seen this before)
'greek character quiet with space' is the wordplay.
'greek character' becomes 'mu' (letter from the Greek alphabet).
'quiet' becomes 'sh' ('sh' is used to tell someone to be quiet).
'with' says to put letters next to each other.
'space' becomes 'room' (enough space or room to do something).
'mu'+'sh'+'room'='MUSHROOM'
'to' is the link.
(Other definitions for mushroom that I've seen before include "that may be edible?" , "Food fungus" , "explode?" , "Increase with remarkable rapidity" , "Proliferate" .)