Some good sources for range-improvement exercises include the studies by Bai Lin, James Stamp, Max Schlossberg, James Thompson, Cat Anderson, Carmine Caruso and several others. What works for one person may not necessarily work for another. Also, many university professors and qualified teachers can offer suggestions for exercises and can help you make a plan for progress. A private teacher is really a necessity for most musicians to progress. Even top-of-the-line professionals often seek advice from other professionals.
This thread is here to aggregate suggestions and advice, as well as collect stories of success and progress to motivate others. Please note that not all advice in this thread will work for you or is from a qualified professional, but such is the nature of the Internet anyway.
My beginnings with the stratosphere began in 10th grade with a band director who was more interested in winning band trophies than what was best for the students. By 11th grade we were playing basically a Maynard Ferguson show. I was blowing a hurricane throw a tiny hole and shoving the mouthpiece down my throat at the same time. How I circumnavigated injury is beyond me. I then started playing gigs in my hometown and blasting G above high C in a honky tonk bar just didn't cut it so I started playing more melodically taking a break from the stratosphere. When I did begin to approach the stratosphere again, I started very softly knowing that I must first play the note before I can add volume to the note. Possibly the biggest influence during this time was Mendez, basic research and a thin book called Lip Flexibilities by Chas Collins. I also changed my mind set. Instead of trying to get a hot compressed sound, I tried to sound more like a piccolo trumpet. Nice and controlled. Now, I can do both. Hot compressed sound and a nice piccolo sound. Not as nice as a piccolo trumpet but I think you get my drift. I treated the very high notes with equality to the other notes. I'm now flirting with a triple C. Often the lips will shut down but that's because I lose control of the wind and probably adding too much mouthpiece pressure. I have no idea where I would use such a note but I saw a video of Rashwan Ross doing it and I thought, "why not"
Also, smaller aperture means less wind. I know it sounds counter intuitive but if the hole is smaller......need I say more?
Here's a video of Rashawn showing Rapa how to do it