The most common and most likely answer to this crossword clue is the 7 letter word NEMESES
I've seen this crossword clue in The Daily Telegraph
N
E
M
E
S
E
S
Buddy explains!

The straight part
We know that "greatest rivals" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case greatest rivals = nemeses.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does NEMESES mean?
NEMESES noun- An archenemy
- (chiefly non-North American usage) A person or character who specifically brings about the downfall of another person or character.
- The principle of retributive justice.
- (usually in the singular) A punishment or defeat that is deserved and cannot be avoided.
- The polar opposite of a character.
- A righteous infliction of retribution manifested by an appropriate agent.