The most common and most likely answer to this crossword clue is the 6 letter word LAXEST
I've seen this crossword clue in The Daily Telegraph
L
A
X
E
S
T
Buddy explains!

The straight part
We know that "most careless" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case most careless = laxest.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 LAXEST mean?
LAXEST adjective- Lenient and allowing for deviation; not strict.
- Loose; not tight or taut.
- Lacking care; neglectful, negligent.
- Having a looseness of the bowels; diarrheal.
- Describing an associative monoidal functor.