The most common and most likely answer to this crossword clue is the 8 letter word JUBILATE
I've seen this crossword clue in The Times
J
U
B
I
L
A
T
E
Buddy explains!

The straight part
We know that "to express great happiness" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case to express great happiness = jubilate.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 JUBILATE mean?
JUBILATE verb- To show elation or triumph; to rejoice.