The most common and most likely answer to this crossword clue is the 6 letter word LOVING
I've seen this crossword clue in The Guardian
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More clues leading to the result LOVING
Buddy explains!

The straight part
We know that "Greatly appreciating" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case greatly appreciating = loving.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 LOVING mean?
LOVING verb- (usually transitive, sometimes intransitive, stative) To have a strong affection for (someone or something).
- To need, thrive on.
- To be strongly inclined towards something; an emphatic form of like.
- (usually transitive, sometimes intransitive) To care deeply about, to be dedicated to (someone or something).
- To derive delight from a fact or situation.
- To have sex with (perhaps from make love).
- To praise; commend.
- To praise as of value; prize; set a price on.
- The action of the verb to love.
- Expressing a large amount of love to other people; affectionate.