Love playing game that's last of series (5)
I believe the answer is:
omega
'last of series' is the definition.
(final letter of the Greek alphabet)
'love playing game' is the wordplay.
'love' becomes 'o' (In tennis, 'love' means 'zero').
'playing' is an anagram indicator.
'game' anagrammed gives 'mega'.
'o'+'mega'='OMEGA'
'that's' is the link.
(Other definitions for omega that I've seen before include "end of series" , "Greek finalist" , "Ending of a series or sequence" , "character in Greek literature?" , "letter from abroad" .)