Well, you aren't doing anything after spawning the process, so the script is ending and killing the process. Try the following:

use Expect;
$Expect::Debug = 1;
my $string= '/usr/bin/ssh 127.0.0.1';
my $exp=Expect->spawn($string) || die "error $!\n";

$exp->expect(2, "password: ");
$exp->send("XXXXXXXX\n");
$exp->expect(2,'$');
$exp->send("ls\n");
$exp->expect(2,'$');
[download]

This works fine on my system.

Update:

If your goal is to interact with the ssh session after scripting the login, try adding:

$exp->interact();
[download]

after you have finished logging in.