Seriously. At least for those of us who are beginners the one magical thing you can do to improve your singing is to practice, practice, practice. It does not even cost one cent.
Using this fabulous method I increased the range of my chest voice by a fourth from E4 to A4 within just two or three weeks. Now even B4 is no problem most of the time.
On the other hand, there seems to be a dead-end at around E5. I can only (and barely) hit that note in whistle-voice and I can not imagine, how I could ever go any higher. So that might just be the physical top of my range. Well, I get down to E2 quite comfortably and three octaves is not bad, I guess.
But the core message is: Practice. There is NO way around it. No vocal training system, no matter how well-advertised it is, can eliminate the necessity of practicing regularly.