The most common and most likely answer to this crossword clue is the 14 letter word SUPERSATURATED
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The straight part
We know that "Getting dangerously full" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case getting dangerously full = supersaturated.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 SUPERSATURATED mean?
SUPERSATURATED verb- To cause a solution to have more solute dissolved in it than it can stably contain at current conditions.
- (of a solution) More concentrated than is normally possible.
- (of a vapor) Having a vapor pressure higher than is normally possible.