The most common and most likely answer to this crossword clue is the 9 letter word ANTIPATER
I've seen this crossword clue in The Guardian
A
N
T
I
P
A
T
E
R
More clues leading to the result ANTIPATER
Buddy explains!
The straight part
We know that "Alexander the Great’s regent" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case alexander the great’s regent = antipater.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.