Is there such a thing as a maximum aperture that a lens can be open to? What about a minimum aperture that it can be closed to? Do these concepts even make any sense? Is there a lens with the narrowest aperture in the world? Is there one with the widest?
Answer
An aperture could be closed which is effectively an infinitely large f-stop number since no light gets through. The fastest possible (smallest f number) is a bit harder. The speed of a lens is limited by the ratio of the entrance pupil to the focal length of the lens. The longer the focal length, the bigger the entrance pupil must be. In theory you could make one very very large, but eventually the amount of glass is going to make it so you physically would lose more light than you were gaining.
There "record" for fastest lens is arguably the f/.33 Super-Q-Gigantar 40mm, but it was really just a marketing gimmick and only one was ever made. It isn't actually functional. There is a functional f/.7 lens of which 10 were made. Six were purchased by Nasa, Carl Zeiss kept one for himself and 3 of them were purchased by Stanley Kubrick and used in the film Barry Lyndon.
In theory, it should be possible to design lenses faster than this, but the cost and benefit are simply not worth it. The lenses become too costly and complex and don't offer any significant benefit for the effort since the difficulty goes up faster than exponentially. (Since each f/stop requires a doubling of the size and physical issues make it more that twice as complicated for each additional f-stop.)
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