The most common and most likely answer to this crossword clue is the 8 letter word STERNEST
I've seen this crossword clue in Independent
S
T
E
R
N
E
S
T
More clues leading to the result STERNEST
- Back in Paris is becoming rigorous in the extreme
- Back head abandoning exam, it’s most rigorous
- Most serious cryptic setters assume name
- Most stony streets in which flier will settle
- Most unfriendly old author on street
- Most unforgiving and most backward?
- Hemingway goes after virtuous person, being most sober
Buddy explains!

The straight part
We know that "most rigorous" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case most rigorous = sternest.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 STERNEST mean?
STERNEST adjective- Having a hardness and severity of nature or manner.
- Grim and forbidding in appearance.