While most photographers prefer fast lenses that range down to f/1.2, are there any lenses that go smaller than f/32? If there are, what is so special about those lenses that an aperture of f/32 cannot do? Rarely do I set it to f/32, because it compromises the whole exposure settings.
In terms of cameras that use 35mm sized or smaller film, or have a sensor no larger than 36mm x 24mm you rarely see a lens with an aperture narrower than f/32.
Medium and large format cameras, on the other hand, frequently use lenses with apertures of f/32 and narrower.
The thing to remember is that an f-number is a ratio, not a specific sized aperture. Also remember that the larger the recording medium, the wider the field of view for a lens of a specific focal length. A 50mm lens, for instance, is considered mildly telephoto on an APS-C camera since it yields a field of view (FoV) equivalent to an 80mm lens on a full frame sensor. That same 50mm lens is considered a standard lens on a 35mm film camera or a full frame digital camera. On a large view camera, a 50mm lens is considered wide angle because of the FoV it yields. As focal lengths must get longer to obtain the same FoV on larger film or sensor sizes compared to lenses for smaller formats, the f/ratio needed to yield the same depth of field (DoF) also goes up.
There was even a group of seven San Francisco based photographers in the 1930s, whose members included Ansel Adams and Edward Weston., named Group f/64 in reference to a common aperture setting used with large format cameras to get great depth of field for landscape photography.
Yes. For example, my sigma 105mm can go up to f/45. Basically, such small aperture can produce a very large DOF, but In most cameras and lenses combination the quality of the image starts to decrease after f/16 because of the diffraction of light, and usually the DOF at f/16 is large enough. I don't know what so special about the f/32 but i guess that in most cases the image quality with smaller aperture will decrease significantly and manufacturers don't see any advantage enabling smaller aperture.
If you're using such small aperture just to get a longer exposure, you should consider using an ND filter which reduce the amount of light. For example a 0.9ND (X3) will reduce 3 stops and that means X8 longer exposure with the same parameters, or same exposure with a bigger aperture.
You can get some pretty good ND filters that only transfer 1/1000 of the given light, thus, creating very long exposures even in the middle of the day.
For further information about diffraction and diffraction limit you may start here
There are, but not many for full frame cameras.
Problem with small apertures is that because of diffraction of light, image can get blurry. However, diffraction is related to absolute width of an opening, and aperture is relative measure, so mostly with medium frame and large frame cameras such apertures make sense.
Similarly, compact cameras which have even smaller sensors, usually max out at f8 or some such.
While I understand that you may want to reduce the amount of light or try to get a deeper depth of field, you should keep in mind that as you close you aperture, you're loosing resolution and sharpness, since the smallest detail your lens can "see" depends on open diameter (aperture).
As a rule of thumb: when you go from a diameter to half of it, the smaller detail your lens can "see" become bigger, actually doubling in size. This is something caused by diffraction.
For example, a 50 mm lens open at f/4, your lens is collecting light from a diameter of 50/4 = 12.5 mm. But when you close your aperture to f/32 you're collecting light from a tiny ~1.56 mm diameter, and everything will blur. Do you really want such blurred image? Most of the time not.
Actually real lenses don't give their sharpest view when used at full aperture because of other optical aberrations that affect sharpness, so reducing aperture often helps sharpness. But usually when you reduce aperture more than f/8 you start to see details going away. So there's no point in building lenses with smaller maximum aperture, as explained also here: