The most common and most likely answer to this crossword clue is the 10 letter word STRETCHING
I've seen this crossword clue in The Daily Telegraph
S
T
R
E
T
C
H
I
N
G
Buddy explains!

The straight part
We know that "Exaggerating" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case exaggerating = stretching.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 STRETCHING mean?
STRETCHING noun- The act by which something is stretched.
- A form of physical exercise in which a specific skeletal muscle (or muscle group) is deliberately elongated to its fullest length in order to improve the muscle's felt elasticity and reaffirm comfortable muscle tone.
- To lengthen by pulling.
- To lengthen when pulled.
- To pull tight.
- To get more use than expected from a limited resource.
- To make inaccurate by exaggeration.
- To extend physically, especially from limit point to limit point.
- To extend one’s limbs or another part of the body in order to improve the elasticity of one's muscles
- To extend to a limit point
- To increase.
- To stretch the truth; to exaggerate.
- To sail by the wind under press of canvas.
- To execute by hanging.
- To make great demands on the capacity or resources of something.
- Requiring a high level of effort or performance.