Greek character's climbing tower, close to exhaustion (7)
I believe the answer is:
upsilon
'greek character's' is the definition.
(letter from the Greek alphabet)
'climbing tower close to exhaustion' is the wordplay.
'climbing' becomes 'up'.
'tower' becomes 'silo' (silo is a kind of tower).
'close to' suggests the final letters (the close of something is the end).
The last letter of 'exhaustion' is 'n'.
'up'+'silo'+'n'='UPSILON'
(Other definitions for upsilon that I've seen before include "Greek character" , "Greek letter" , "Character from abroad" , "Greek 'u'" .)