The most common and most likely answer to this crossword clue is the 11 letter word SINGULARITY
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Buddy explains!

The straight part
We know that "Oddity" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case oddity = singularity.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.
Crossword Answer definitions
SINGULARITY noun- The state of being singular, distinct, peculiar, uncommon or unusual
- A point where all parallel lines meet
- A point where a measured variable reaches unmeasurable or infinite value
- The value or range of values of a function for which a derivative does not exist
- A point or region in spacetime in which gravitational forces cause matter to have an infinite density; associated with black holes
- A proposed point in the technological future at which artificial intelligences become capable of augmenting and improving themselves, leading to an explosive growth in intelligence.
- Anything singular, rare, or curious.
- Possession of a particular or exclusive privilege, prerogative, or distinction.
- Celibacy