The most common and most likely answer to this crossword clue is the 4 plus 4 plus 9 letter word BEND OVER BACKWARDS
I've seen this crossword clue in The Times
B
E
N
D
O
V
E
R
B
A
C
K
W
A
R
D
S
Buddy explains!
The straight part
We know that "Try one’s hardest" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case try one’s hardest = bend over backwards.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.