The most common and most likely answer to this crossword clue is the 9 plus 7 letter word INGROWING TOENAIL
I've seen this crossword clue in The Guardian
I
N
G
R
O
W
I
N
G
T
O
E
N
A
I
L
Buddy explains!
The straight part
We know that "Extreme agony" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case extreme agony = ingrowing toenail.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.