The most common and most likely answer to this crossword clue is the 10 letter word SENESCENCE
I've seen this crossword clue in The Guardian
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More clues leading to the result SENESCENCE
Buddy explains!

The straight part
We know that "process of aging" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case process of aging = senescence.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 SENESCENCE mean?
SENESCENCE noun- The state or process of ageing, especially in humans; old age.
- Ceasing to divide by mitosis because of shortening of telomeres or excessive DNA damage.
- Old age; accumulated damage to macromolecules, cells, tissues and organs with the passage of time.
- Fruit senescence, leading to ripening of fruit.
- (of a cell) Condition when the cell ceases to divide.