lens - Why do zoom lenses and compact cameras have varied maximum aperture across the zoom range?

Why does a camera's maximum (allowed) aperture get smaller when you increase its zoom?

Answer

The short answer is because it is cheaper to manufacture such lenses. The longer the lens and the wider the aperture, the larger the optical elements in the lens - thus larger the expense to produce them.

A lens like 70-200/2.8 must have a front optical element of 200mm/2.8=72mm, which is quite a chunk of glass. On the other hand, the 70-300/4-5.6 needs to be 300mm/5.6=54mm wide. If it were f/4 through its full range, the optical element would need to be 75mm wide - even larger than the much more expensive 70-200/2.8.

In your question, you say "the camera's maximum aperture". The camera does not have an aperture - the lens does. Minor but important difference, especially for SLRs - once you remove the lens you see that the camera is just a light bucket with a big hole in the front.

DETAILS:

The aperture is the ratio of the focal length of the lens to the size of the front optical element. Essentially

aperture = focal length / optical element size

For example, a 50mm f/1.8 lens has a 28 mm (50/1.8) element size.

If you're wondering why the f-stop numbers don't seem to be linear (they're not), it is because the amount of light collected by the lens is proportional to the focal length divided by the aperture squared. Because of this power of 2, f/4 collects twice as much light as f/5.6, since 5.6/4=sqrt(2).