quotemeta 
$constant = quotemeta '$$';
$line_count =0;
while (<FILE>){
  if($constant ne "") {
     if($_ =~ m/^($constant)/) {
       $line_count++;
     }
  }
}

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UPDATE: Geez... you asked this question 2 days ago, and got the answer --

In reply to Re: Issue with regex matching by toolic
in thread Reaped: Issue with regex matching by NodeReaper

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