The most common and most likely answer to this crossword clue is the 7 letter word NOSIEST
I've seen this crossword clue in The Guardian
N
O
S
I
E
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T
More clues leading to the result NOSIEST
- Case of Navajo and the sleep that never ends? Most curious!
- Like the most interfering person, giving one no rest for the most part
- Lack of rest breaks when leaving — most curious
- Absence of nap, having start of afternoon off, is most curious
- Most inquisitive when without sleep? Not quite!
- Most curious, spiralling section of Lhotse is onscreen
- Rolling Stone is most meddlesome
Buddy explains!

The straight part
We know that "most curious" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case most curious = nosiest.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 NOSIEST mean?
NOSIEST adjective- Prying, inquisitive or curious in other’s affairs; tending to snoop or meddle.
- Having a large or elongated nose.