All right, here's my attempt:

UPDATE: fixed it to account for "aba" palindromes, thanks BrowserUk:

UPDATE2: fixed again, thanks ccn:

#!/usr/bin/perl -w
use strict;

my $input = <STDIN>;
chomp($input);

my $longest_palindrome = '';

# look for two occurrences of the same character back to back or with 
+another
# character in-between them to find palindromes:
while ($input =~ /((.).?\2)/g)
{
    # the first character beyond the three or two character middle of 
+a
    # palindrome:
    my $match_position = pos($input);

    # get the positions of the two matching characters:
    my $left_pos  = $match_position - length $1;
    my $right_pos = $match_position - 1;

    # now go looking to the left and right of each matching character
    # for more matching characters:
    while (nextCharactersMatch($input, $left_pos, $right_pos))
    {
        last if ($left_pos <= 0 or $right_pos >= length $input);

        $left_pos--;
        $right_pos++;
    }

    # extract the palindrome:
    my $offset     = $left_pos;
    my $length     = ($right_pos - $left_pos) + 1;
    my $palindrome = substr($input, $offset, $length);

    $longest_palindrome = $palindrome
        if ($length > length $longest_palindrome);

    # backtrack, to find palindromes within this palindrome:
    pos($input) -= (length($1) - 1);
}

print $longest_palindrome;





sub nextCharactersMatch
{
    my ($input, $left_pos, $right_pos) = @_;

    return 1 if (substr($input, $left_pos - 1, 1)
            eq substr($input, $right_pos + 1, 1));
}
[download]