The most common and most likely answer to this crossword clue is the 7 letter word RASHEST
I've seen this crossword clue in The Daily Telegraph
R
A
S
H
E
S
T
Buddy explains!

The straight part
We know that "Most indiscreet" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case most indiscreet = rashest.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 RASHEST mean?
RASHEST adjective- Acting too quickly without considering the risks and consequences; not careful; hasty.
- So dry as to fall out of the ear with handling, as corn.
- Requiring sudden action; pressing; urgent.
- Fast-acting.