GSD has an succinct but accurate view of the whole business. I'm not a lens designer, but I have studied lens design and I find the diagrams interesting because it shows me how the designer attacked the problems. There are many ways to "skin the cat."
As a rule of thumb, simplicity is king. Whenever you add elements to correct something, you always add more issues to correct. Basically, the trick is to get it done with as much simplicity as possible. Someone mentioned aspheric elements above. That is an example of using simplicity because aspheric elements can replace several traditional elements.
Classic 35mm designs like the Nikkor 105 f2.5 or the Leica 90mm f2.8 are simple - just a handful of elements. The problem today is that people want zoom lenses and therein lies the difficulty for the lens designer. It is hard enough to correct a single focal length - it is a matter of selecting compromises to correct one with a range of focal lengths. So the zoom lenses don't have more elements because they are better. They have more elements because they are more complicated. The complication shows up as more compromise in the design and performance of the lens. It is a tradeoff between flexibility and performance. When I see that a lens has 15 elements I scratch my head and wonder that the thing can make photographs at all. It can, of course, but it takes a computer to do all the calculations that go into making one of these monsters.
The very highest performing lenses are those that do not need to zoom and do not need to focus (focusing is handled by the camera rather than the lens itself.) Examples are view camera lenses and others like the Mamiya RZ fixed lenses which are really short view camera lenses with a smaller image circle. These lenses do not need to zoom or focus. Nothing moves inside the lens barrel. They just need to project a well corrected, sharp image and they do. They are very simple compared to 35mm sized zoom lenses. You simply can't believe what they can do until you use one.
So don't view the complement of elements as a mark of quality or performance, view it as the mark of complexity and the designer's desire to lessen the impact of the compromises.