With slight changes, this can be made tail-recursive, which means the recursion can be elimited completely.

Tail-recursive variant:

sub cleverGuess {
   my( $lower, $higher ) = @_;

   my $guess = int(($lower + $higher)/2);
   print "Guessing: $guess\n";

   if ($guess == $ans) {
      print "The guess was correct!";
      return;
   }

   elsif ($ans < $guess) {
      print "Lower...";
      $higher = $guess-1;
   }

   else {
      print "Higher...";
      $lower = $guess + 1;
   }

   cleverGuess($lower, $higher);
}
[download]

Tail-recursion replaced with a loop:

sub cleverGuess {
   my( $lower, $higher ) = @_;

   for (;;) {
      my $guess = int(($lower + $higher)/2);
      print "Guessing: $guess\n";

      if ($guess == $ans) {
         print "The guess was correct!";
         return;
      }

      elsif ($ans < $guess) {
         print "Lower...";
         $higher = $guess-1;
      }

      else {
         print "Higher...";
         $lower = $guess + 1;
      }
   }
}
[download]

I don't mean to diminish your effort. I just wish to illustrate that power of recursion is sometimes an illusion. Performance should increase by removing recursion where it's not needed, and now you know how!