The most common and most likely answer to this crossword clue is the 13 letter word INFINITESIMAL
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More clues leading to the result INFINITESIMAL
Buddy explains!

The straight part
We know that "very insignificant" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case very insignificant = infinitesimal.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 INFINITESIMAL mean?
INFINITESIMAL noun- A non-zero quantity whose magnitude is smaller than any positive number (by definition it is not a real number).
- Incalculably, exceedingly, or immeasurably minute; vanishingly small.
- Of or pertaining to values that approach zero as a limit.
- Very small.