Orbit is not about going high. The Space Station is only about 400 km up. Orbit is about going sideways very fast: around 7.8 km/s, or 28,000 km/h. Getting a rocket to that speed turns out to be a battle against one stubborn piece of arithmetic.
A rocket moves by throwing mass backwards. The more fuel it burns, the lighter it gets, so each kilogram of fuel speeds up a lighter and lighter rocket. The result is the Tsiolkovsky rocket equation, published in 1903:
speed gained = exhaust speed × ln(full mass ÷ empty mass)
The "ln" is a natural logarithm, and it is the villain. To double your speed gain you do not double the fuel. You square the full-to-empty mass ratio. Each extra bit of speed costs more fuel than the last.
Real engines throw their exhaust out at roughly 3 to 4.5 km/s, depending on the fuel. Reaching orbit needs a speed gain of about 9 to 10 km/s: the 7.8 km/s of orbital speed plus another 1.5 to 2 km/s lost to gravity pulling the rocket back during the climb, steering, and air drag. So the rocket must be mostly fuel. A common textbook example, with a 9.7 km/s requirement and a 4.5 km/s exhaust, needs about 88% of a single-stage rocket's launch mass to be propellant.
Below, the rocket carries a payload and is built from equal stages. Each stage's tanks and engines weigh a fixed share of the stage's fuelled weight. Slide the controls and watch the speed gain against the orbit line.
Orbit line drawn at 9.4 km/s, the middle of the 9 to 10 km/s range. This is the ideal equation: real launches lose some more to air and gravity, and real tanks are never exactly 10% of a stage.
Start with one stage and 10% structure. A single stage that is 90% fuel gets a speed gain of about 7.3 km/s with a 2% payload: short of orbit. Pushing the structure slider down helps, but the maths has a ceiling. With tanks and engines at 10% of the stage, even a rocket carrying no payload at all could reach only the exhaust speed times ln 10, which is about 7.8 km/s on kerosene-type fuel. That is barely the orbital speed itself, before any losses, so a single stage like this is out.
Now add a second stage. The first stage gets dropped when its fuel is gone, so the rocket stops hauling empty tanks. Two stages already clear the orbit line in the model. A third stage adds speed gain with a smaller payload cost, and the gains flatten out after that, because every stage also adds its own engines and tanks.
Switching to hydrogen-type exhaust raises every bar by almost 30%. In practice hydrogen is so fluffy that its tanks are bulkier, so try nudging the structure slider up when you switch.
The Saturn V that sent Apollo to the Moon had three stages. Most orbital rockets today have two. Rockets that reuse their first stage are still staged: the idea is the same, only the dropped part now comes back. The rocket equation is why launch vehicles look the way they do: tall stacks of tanks, mostly fuel, throwing away the heavy parts as soon as they are empty.
It is also why a payload is so small. Even with two stages, a rocket like the one in the demo carries only a few percent of its launch mass as cargo. All the rest is the price of speed.