The most common and most likely answer to this crossword clue is the 9 letter word UTTERMOST
I've seen this crossword clue in The Times
U
T
T
E
R
M
O
S
T
More clues leading to the result UTTERMOST
- Voice witty remark about second last
- What one with the biggest mouth does is extreme
- Extreme talk supported by a majority
- Say more than anyone else, being extreme
- Extreme way to dominate as speaker?
- –Highest possible, say, majority
- Make the speech that’s longest and greatest
- Mutters to bum as much as possible
- What the keenest do to win a speaking competition?
- Say more than anyone else— to the greatest degree
Buddy explains!

The straight part
We know that "greatest" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case greatest = uttermost.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 UTTERMOST mean?
UTTERMOST noun- The utmost; the highest or greatest degree; the farthest extent.
- Outermost.
- Extreme; utmost; of the farthest, greatest, or highest degree.