The most common and most likely answer to this crossword clue is the 6 plus 8 letter word DEEPLY OFFENDED
I've seen this crossword clue in The Guardian
D
E
E
P
L
Y
O
F
F
E
N
D
E
D
Buddy explains!
The straight part
We know that "most indignant" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case most indignant = deeply offended.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.