In response to your updated question,

my $name = '(\w[\w-]*\w?)';       # any valid hostname segment
my $nnam = "(($name\\.)*$name)";  # one or more "name" segments

my %domainnames = (
  "$nname\\.com" => 'a commercial domain',
  "$nname\\.edu" => 'an educational institution',
);
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is a solution. When quoting that with double quotes, you need to escape $ (unless you want interpolation), @ (unless you want interpolation), " and \ by preceeding them with \, so the regexp is $nname\.com would be "$nname\\.com" as a string literal.

But it's simpler to use qr/.../ (like I suggested in my earlier post):

my $name = qr/(\w[\w-]*\w?)/;      # any valid hostname segment
my $nnam = qr/(($name\.)*$name)/;  # one or more "name" segments

my %domainnames = (
  qr/$nname\.com/ => 'a commercial domain',
  qr/$nname\.edu/ => 'an educational institution',
);
[download]

However, it's a big waste to stringify a compiled regexp, so you might want to alter your structure for a major speed boost. I doubt you actually need a hash, so here's a solution using an array:

my $name = qr/(\w[\w-]*\w?)/;      # any valid hostname segment
my $nnam = qr/(($name\.)*$name)/;  # one or more "name" segments

my @domainnames = (
  { re => qr/$nname\.com/, desc => 'a commercial domain'        },
  { re => qr/$nname\.edu/, desc => 'an educational institution' },
);
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If you really do need a hash:

my $name = qr/(\w[\w-]*\w?)/;      # any valid hostname segment
my $nnam = qr/(($name\.)*$name)/;  # one or more "name" segments

my @domainnames = (
  qr/$nname\.com/ => 'a commercial domain',
  qr/$nname\.edu/ => 'an educational institution',
);

my %domainnames;
foreach (0 .. @domainnames/2-1) {
   $domainnames{$domainnames[$_*2+0]} = {
      re   => $domainnames[$_*2+0],
      desc => $domainnames[$_*2+1],
   };
}
[download]