Context-free grammar for"not-at-all" palindromes
Just use something like:

$
S \rightarrow a S b \mid a S c \mid b S a \mid b S c \mid c S a \mid c S b \mid a b \mid a c \mid b a \mid b c \mid c a \mid c b
$

This ensures the $i$-th symbol from the beginning is different from the $i$-th from the end.