The most common and most likely answer to this crossword clue is the 2 plus 2 plus 3 plus 6 letter word UP TO THE ELBOWS
I've seen this crossword clue in Independent
U
P
T
O
T
H
E
E
L
B
O
W
S
Buddy explains!
The straight part
We know that "Completely engrossed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case completely engrossed = up to the elbows.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.