The most common and most likely answer to this crossword clue is the 6 plus 1 plus 4 letter word GEORGE W BUSH
I've seen this crossword clue in The Guardian
G
E
O
R
G
E
W
B
U
S
H
Buddy explains!
The straight part
We know that "Silly bugger whose next-but-one successor was worse" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case silly bugger whose next-but-one successor was worse = george w bush.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.