The most common and most likely answer to this crossword clue is the 8 letter word PRICIEST
I've seen this crossword clue in The Daily Telegraph
P
R
I
C
I
E
S
T
More clues leading to the result PRICIEST
Buddy explains!

The straight part
We know that "extremely dear" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case extremely dear = priciest.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 PRICIEST mean?
PRICIEST adjective- Expensive, dear.