First, you should be using a placeholder in the query string, and preparing the query statement outside the foreach loop. Second, you can either include a "LIMIT 2" clause at the end of the query string, as suggested above, or you can simply exit the inner "fetch" loop after doing two rows (but using the LIMIT clause would probably be more efficient, assuming your database server supports it).

Regarding the placeholder usage, it would go like this (and I'm throwing in both the LIMIT clause and a counter for the inner fetch loop):

my $sql = "SELECT G.Region, G.Score FROM Gov_regions_scores_TEMP G, Re
+gion_lookup R
  WHERE R.String = ? AND RTRIM(R.Place) = RTRIM(G.Region) LIMIT 2";
my $sth = $dbh->prepare( $sql );

foreach ( @regions_A_array ) {
    $sth->execute( $_ );
    for ( 1, 2 ) {
        my @regions_A = $sth->fetchrow_array;
        print OUTPUT "@regions_A\n";
    }
}
$sth->finish();
[download]

I don't have a means for testing that, but it should be pretty close to what you want. (If it doesn't work, show us exactly how you tried it, what error messages and output you got, if any, and what output you expected.)