Breakthroughs in craftsmanship, not fundimental laws of physics, are what is driving the current space privitization effort. We still use chemical fuel and Newton's Third Law to make a rocket work — don't expect that to change. There is no magic-bean powered carpet in forseeable future. It is improvements in tools and materials handling that is ongoing, and that also allows the creation of better tools as well as final goods, so it has the potential for exponential growth.
Private companies are trying to make space affordable by making vecicles lighter and more cheaply.
The entire launch vehicle and engines are just a tax on top of what you really wanted to orbit. If you could get the weight down to a feather, you still have the weight of the payload and fuel for that weight. So there is a limit.
However, what allows colleges and startups to access space is the microsat. Even when it still costs \$8600 per pound (which comes to $75,000 for a gallon of milk), a useful payload can be built that is only a few pounds.
Look at an Android smartphone: a few ounces for more compuer power than existed onnthe planet during the Space Race, sensors, optics, and communications capability.
Further improvement that could be at the levelmof a breakthrough in commercial availability of orbital experiments (not in physics) would be the development in infrastructure. The microsat is bloated by shielding, more powerful radio, and the need for self-contained power, so it winds up being 10×10×40 cm rather than a smartphone.
A commercial service could offer orbital hosting, where your microsat is placed in a shielded enclosure with a network connection and power, supplied by them. The suite with its heavy shielding does not need to get sent up every time the interesting guts gets replaced! That would reduce costs by an order of magnitude, all, without needing to discover new laws of physics or invent whole new technologies. Its just investment and building on continuing work.
What is happening: smaller stuff. Nanotechnology, spintronics, novel materials, etc.
What is not going to happen, because we know how it works and that there are limits: chemical energy won't be stored any denser than what is already found in high explosives. That is, rocket fuel won't get orders of magnitude more powerful.
(Non-chemical rockets like ion drives are not useful for getting into orbit. Thrust less than 1g is not going to get anywhere.)
So what about replacing launch rockets with something that does not carry the fuel with it? Cannon launchers have limits, and unless you build a many-miles-long railgun it won't help. Getting as much speed as you can while still in the atmosphere means you can save on reaction mass and get oxygen without carrying it all, but how much does that save? It still relies on the compound interest of the rocket equasion and cutting off the bottom saves many times the direct savings. Easy recovery and reuse of the stage (as it's a plane) needs to be weighed against the weight savings of making it disposable!
So what if you wanted to go to the moon in person? Hundreds of pounds with life-support and snacks, comes to millions of dollars.
If you can't get past chemical launch rockets until much later when space industry is already mature, how about making the rocket fuel cheap?
You can't have exotic toxic materials, just as cars are subject to emmission regulation now. The rocket exhaust will need to be eco-friendly, meaning not as powerful as it might be if you could choose the most potent chemicals without constraint.
But lack of rare metals and such will also make it cheap to produce. In a future with advanced chemestry and nanotechnology, potent molecules can be designed and contained safely, using only common life elements. Hey, it could grow on trees. Ultimately, if gathering and arranging the atoms is cheap, you are limited by the amount of energy you are storing. Less power per mass pays penalty exponentially via the rocket equasion, but look at what petrolium is: organic molecules. Or, liquid oxygen and hydrogen: what if that was cheap, other than the cost of energy?
Later, but before the possibility of space elevators, a commercial capability breakthrough will be to source industrial materials off-earth rather than lifting it. We will be importing raw material and manufacured goods, and leaving will be expensive. Since you only need to get to LEO and then all the life-support and living quarters are from space rather than lifted from Earth, my ticket out (a 250 pound man) will be $251,000 with today's rockets. Earning power will increase, fixed infrastrucure cost will be amortized (don't need to buy the airplane when I fly coach), and cost to produce the fuel will be smaller relative to the total consumption per person of energy. Whether that means affordable as saving up for a few years, or cheap, depends on breakthroughs in our energy economy. Cheap practical fusion? Or just nearly-free solar power?