I’m disappointed that Wikipedia still lists the “six simple machines”: Modern elementary textbooks have (thankfully) finally moved on from that list.
The old-fashioned list of six really does only include three, but not he way you’re breaking it down: A wheel (in the sense that it’s a machine) is just a kind of lever, but a pulley really is a distinct one, independent of the wheel.
The way you make a machine out of a wheel is by putting multiple wheels on the same axle, of different radii. But that just means that you’ve got a fulcrum at the axle, and different lengths of arm from that fulcrum. The only thing that makes it different from a lever is that you’re continually replenishing it: As one arm moves out of position, another arm moves in.
A pulley, meanwhile, could be made without wheels. Take low-friction materials, like nylon rope and a block of Teflon with grooves cut in it, and you could make a block and tackle without anything in it that could be described as a wheel. And it would still have the same mechanical advantage as a wheel-based pulley. It’d be impractical, because it’d have more friction, but it’d still work.
And there are many, many other simple machines. A hammerhead is another one. Yes, a hammer usually combines a hammerhead with a handle (which is a lever), but the lever isn’t the only part of the hammer providing mechanical advantage. To prove this: First try to drive a nail into a piece of wood with just your fist, and then pick up a decent-sized rock, and use that to drive the nail into the wood. You’ve got the same lever in both cases, your arm. But the rock is a primitive hammerhead, and that’s what makes the difference.
A sample list of simple machines I can think of off the top of my head (all of which are distinct, and any one of which can be made without using any of the others, hence “simple”):
*An inclined plane
*A block and tackle
*An oblique line
*A hydraulic ram
*A generator-motor pair
I don’t claim that this is a complete list; others can probably think of more. But it amply illustrates that the standard list of six is incomplete.