The most common and most likely answer to this crossword clue is the 4 letter word NECK
I've seen this crossword clue in The Times
N
E
C
K
Buddy explains!
The straight part
We know that "Deeply involved, up to this/kiss" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case deeply involved, up to this/kiss = neck.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 NECK mean?
NECK noun- The part of the body connecting the head and the trunk found in humans and some animals.
- The corresponding part in some other anatomical contexts.
- The part of a shirt, dress etc., which fits a person's neck.
- The tapered part of a bottle toward the opening.
- The slender tubelike extension atop an archegonium, through which the sperm swim to reach the egg.
- The extension of any stringed instrument on which a fingerboard is mounted
- A long narrow tract of land projecting from the main body, or a narrow tract connecting two larger tracts.
- A reduction in size near the end of an object, formed by a groove around it.
- The constriction between the root and crown of a tooth.
- The gorgerin of a capital.
- The small part of a gun between the chase and the swell of the muzzle.
- A person's life.
- A falsehood; a lie.
- Shapeshifting water spirits in Germanic mythology and folklore
- To hang by the neck; strangle; kill, eliminate
- To make love; to intently kiss or cuddle; to canoodle.
- To drink rapidly.
- To decrease in diameter.