The most common and most likely answer to this crossword clue is the 3 plus 3 plus 2 letter word RUN OUT OF
I've seen this crossword clue in The Guardian
R
U
N
O
U
T
O
F
More clues leading to the result RUN OUT OF
Buddy explains!
The straight part
We know that "Fully exhaust" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fully exhaust = run out of.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.