Thanks for those clarifications. Athanasius has supplied a clear explanation the issue of the brackets and you've provided the extra information that $constant need only be counted at the start of the line. With that in mind we can use the simple (but potentially very inefficient) method with index thus:

$constant = '$$';
$line_count =0;
while (<FILE>){
  $line_count++ unless index ($_, $constant);
}
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