As a child, we are taught that there are only a few simple machines and that most others are derived from them. I think I was first exposed to this concept in the third grade. Those are undeniably important, but I think some are really just variations on others, and there are a few that are missing.
Inclined plane. You can use this to use a smaller amount of force to achieve a larger amount of force. For example, a ramp, a stairway, etc., are all examples of inclined plane.
Lever. This is a different way of achieving something similar. Give me a long and strong enough lever and a place to stand, and I can move the earth, said Archimedes. He also needed a fulcrum.
Wheel with axle. This supports a weight and holds the wheel in place under it, while reducing friction and wear at the axle.
Pulley. The earliest pulleys were blocks of wood with a hole for a rope to pass through, and can be used to change the direction (lead) of a rope or give mechanical advantage. This style of pulley gives us the traditional name for a pulley on a boat, a "block". Modern pulleys have a wheel with axle to reduce friction and wear, but they're still the same concept. On boats, they are still called "blocks".
The rest on the traditional list are derivations of these:
The wedge is an application of the inclined plane, intended to split two things. Knives, wedges and several other common tools use this variant.
The screw is another application of the inclined plane, where the plane is twisted around a shaft.
There are other simple machines which deserve mention on the main list.
The wick. A wick is an object, most often a treated string, that uses capillary action to move a liquid, often against the force of gravity. Most commonly this is melted wax, such as in a candle, but there are many other variations. The treatment is something that makes the string fire resistant, such as salt or borax, although sometimes the wick is made of something not flammable in the first place, such as copper or fiberglass. The invention of the wick is lost in pre-history but nearly every culture used it--whether knowledge spread or was independently rediscovered is unknown. Capillary action was not understood to be the underlying mechanism until relatively recently. The first recorded observation of it was by Leonardo.
The gear. This is derived from the wheel and pulley but the idea of using a toothed wheel to transfer force, and gain or lose mechanical advantage seems to date from a few hundred years BC. While the ancient greeks definitely had gears and used them, the earliest engineers to use them on a major scale were the Romans, working in the 2nd century AD.
Archimedes was the earliest engineer we know of to design using gears but we suspect he didn't invent them. The Antikythera mechanism is the earliest surviving geared mechanism, dating from the 2nd century BC. It is a remarkably sophisticated computer for calculating the motion of the planets, the seasons, and other events. It may have been designed by Archimedes himself, but we don't know. It was discovered in an ancient shipwreck in 1901.
The waterwheel and Fluid dynamics. The ancient greeks seem to have invented the water wheel, but again, the Romans were the first to really exploit them. They are really a combination of other simple machines, but they're important enough to deserve mention. They had a number of variations, some using gravity, some using the force of the moving water. In the 12th century, the dutch began using windmills, which are really the same thing, except using fluid dynamic lift instead of drag. It wasn't until the 18th century that it was realized that this (called a "turbine") was the best way of doing it for all types of fluids. The math was worked out by Daniel Bernoulli by 1738, but they really didn't catch on until the mid 19th century. Traditional water wheels, including paddle wheel steam ships, were very common until that time. A famous experiment was conducted in 1845 by Isambard Kingdom Brunel where he arranged a tug of war between two otherwise identical ships, one with paddlewheels and one with a screw propeller, and the screw prop won very convincingly, being able to pull the paddlewheeler against the current on the Thames river.
