Slip-on shoe ultimately designed for Greek character (7)
I believe the answer is:
epsilon
'greek character' is the definition.
(letter from the Greek alphabet)
'slip-on shoe ultimately designed' is the wordplay.
'ultimately' suggests the final letters.
'designed' is an anagram indicator (designed with the letters).
The final letter of 'shoe' is 'e'.
'slip'+'on'+'e'='slipone'
'slipone' anagrammed gives 'EPSILON'.
'for' acts as a link.
(Other definitions for epsilon that I've seen before include "foreign letter" , "character in Greece" , "Greek letter E" , "Greek character" , "Fifth brightest star in a constellation" .)