The most common and most likely answer to this crossword clue is the 8 letter word GIANTISM
I've seen this crossword clue in Financial Times
G
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Buddy explains!
The straight part
We know that "Greatness" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case greatness = giantism.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 GIANTISM mean?
GIANTISM noun- The quality or state of being gigantic; being of abnormally large size.
- A condition where there is over-production of growth hormone by the pituitary gland in a child before the bone growth plates close, resulting in excessive long bone growth, accompanied by muscular weakness and sexual impotence.