What about some alternative ways to lift stuff into LEO and explore space? I'll start.
The main consideration here is, that conventional multi-stage rockets are mostly good in lifting all the fuel they burn and the engines and fuel tanks needed to do that into orbit. Payload is at best around 6% of the total weight, and they scale very badly.
1) Like Guden Oden said: nucleair propulsion. It works and has been tested. The main disadvantages are the lack of shielding you want (radiation danger for the engineers and flight crew on the ground) and the radioactive waste that comes back down. And that you have to run them close to critical. But they work really well. Although, perhaps only useable if very spectacular faillures are an option.
2) SCRAMjets. They are tested and work, but due to the very small tolerances to the air flow through the engine and the very high stresses and temperatures you have to take faillure into account. But the performance is very promising, and they're easily converted to rocket flight when you get high enough.
3) SSTO (Single Stage To Orbit) rockets. They are tested and work, even better than most alternatives. The only disadvantage I can think of is that they scale pretty bad. Good for something up to a metric ton or so. Due to their simplicity, this will probably be the safest and cheapest option.
4) Lifters (ion drives). They are tested on small scales and work. Make a clear, horizontal aluminium foil honey comb, and put an isolated grid of wires on the top. Power it by beaming microwaves at it. The high voltage created will accelerate the air downwards. Main disadvances: never tested on a large scale, and you need large structures to lift large masses. Probably only useful to a weight of around a few metric tons, that would have to accelerate into orbit from the top of the atmosphere. But extremely cheap.
5) Large, heated hydrogen balloons. They are tested and work. Carry their fuel inside. The main disadvantage is, that I doubt you'll get much higher than about 50 kilometers in the very best case, and so need rocket power for the most part anyway.
6) Microwave powered engines. They are tested on a small scale with different mechanisms to heat the stuff and they work. Take a plane, put a large microwave receiver on the bottom (wire grid embedded as the honeycomb metal in composites), and heat air (first stage) or some electrolytic liquid (second stage) with continuous massive discharges through it, and accelerating the plasma. Main disadvantages: it's big. Especially the transmitter. And you lose all power when the beam loses sight of the receiver. But you only need large amounts of electricity to power it, and the weight it can lift is mostly limited by the energy you can beam upwards. This is for the really big stuff.
7) Laser powered engines. They are tested on a small scale and work. They work like a jet or rocket engine, in which you shine a high-powered laser from the back, directly into the combustion chamber, and use that to heat the air or other reaction mass. Main disadvantages: you need a really good way to keep the laser on focus, and they have bad efficiency, as the laser itself only has one of at most a few percent. Probably not very useful, unless we can come up with better lasers.
8) A big gas gun. They are tested and work. Take a very long barrel (probably made of composites), and pump gas in by combustion every few meters behind the projectile. And fire that projectile at the sky. Main disadvantages: It's very big; you need a fair bit of mountain for the best result, and you need a very high speed to counter the air friction throughout the remaining atmosphere. But it's very simple, operational costs are very low and it works great for dense payloads of small and medium weight. No delicate stuff, though.
9) A lineair accelerator. They are tested and work. Take a very long tube and add coils around every meter or so. Put something covered with metal inside (aluminium would work well), and accelerate it to high speed. It works like a gas gun, but you have more control over the acceleration and it scales much better. Main disadvantages: same as with gas guns. Use this for dense bulk goods, and for things that carry their own rocket engine to help counter the air friction. And a small application is using it as a thruster that shoots metal pellets at very high speed.
10) Orion. Use nuclear explosions (fission or fusion) to power your rocket engine. They aren't tested (AFAIK), but would very likely work extremely well (MUCH better than anything else), if you can keep your ship in one piece. There are two different approaches to this, depending on the scale of the explosions. The first one is like this: take a very thick buffer plate, attach massive springs and dampers, and layer those with lots and lots of other stuff to dampen the shock, throw a nuke underneath and blow it. The other variant uses an engine that consists of a chamber in which you shoot small pellets of He3/H2, embedded in glass, and have them fuse by beaming powerful lasers on it. And a third variant is like a combination of both, but uses anti-matter to power the explosion. For brute force, nothing beats Orion. And it would work everywhere, and scale extremely well. Nothing is too heavy to lift. But the practicality of it is another matter. Let alone fireing lots of nukes inside the atmosphere to launch it.
11) Ion drives. They are tested and work. They work by accelerating very little mass (an ion stream) to a significant part of the speed of light (some even very close to that). They trade mass for speed. They range from tiny to very bulky. Their main disadvantages: the thrust is very small. But they can go on almost indefinitely, and over time they can reach extremely high speeds. Good for the patient, for vector corrections and small, unmanned space craft that need a very long operational lifespan.
12) Fusion drives. They aren't tested as such, but all parts that make them work are. Still, it would be quite an engineering challenge to put them together. They function by creating a hot plasma like in a Tokamak fusion reactor, and compressing that electromagnetically to a very hot and narrow beam, at which point fusion occurs. That speeds up the plasma tremedously, and generates a lot of radiation and particles that can be used to generate electricity, as with a Tokamak. The main disadvantages are the difficulty to make that happen and keeping that very hot plasma away from it's enclosure, and the length of the tube needed to do all that. Think hundreds of meters, or even in excess of a kilometer. Still, they're probably your best bet for serious space exploration.
13) Bussard Ramscoops. They aren't tested and might not work, but they offer all the benefits of a fusion drive and unlimited fuel. They're basically a fusion drive that generates two very strong electromagnetic fields, that charge and scoop up the small amount of hydrogen available even in empty space. That would require quite some astonishing engineering challenges, and the fusion reaction would be pretty hard to accomplish with simple hydrogen, instead of He3/H2/H3. Still, while a very old idea, it's still the best one we have to conquer interstellar space.
14) A solar sail. This is tested and works. You just put a large and very thin sheet of metallized plastic into orbit, and it will be pushed away from the Sun through the impact of charged particles (the solar wind). The main disadvantages are the bulk and strength of the material needed, and that you mostly only can get away from the Sun. This would even allow for interstellar flight. The acceleration would be small, but it builds.
15) The Moon (or a moon, or very large asteroid that consists mostly of metal). This is not tested, but would surely work. Take a very large body, drill a hole through the center, install a linear accelerator in that tube, mine and fire most of it's mass through at a fair bit of the speed of light. The main disadvantages are, that it requires engineering on an extremely massive scale, and that it's single-use only. But with that, you can go anywhere inside our galaxy within two years relative time. For the impatient who have massive armies of intelligent robotic workers.
