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?
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.)
Physics plays a role in answering your question and that information is out there. The basics from that linked discussion are that the index of refraction of the lens material will affect the maximum aperture you can achieve, so for pure glass that has an index of refraction of 1.5, the maximum aperture would be f/0.5 or thereabouts. Better substances, such as diamonds, with an index of refraction of 2.417 can give you an aperture of f/0.235 with a corresponding insane cost of ownership (consider just how much a lens of pure diamond might cost). The lensmaker's equation is the basis for the numbers.
As for minimum aperture, you could basically get down to what amounts to an atomic size hole, large enough for one photon to pass through, but that's useless for, well, anything. For a lot of lenses, getting to some place around f/11 or higher results in loss of sharpness as a function of diffraction, so f/32 is about the top out point for 35mm lenses though they can get smaller for larger formats and do so. Pinhole lenses are often in the smaller range, as much as f/177 (Lensbaby has one like this). Still, even if the optics were perfectly able to handle something like this, consider what the ISO and shutter speeds would need to be to get an image, so at some point, the value of this is pretty much zero unless you're into blurry abstracts.
There are many terms related to aperture, but let's get the most interesting to us: after wikipedia: "angular aperture N of a lens is expressed by the f-number, written f/, which is the ratio of the focal length f to the diameter of the entrance pupil D:"
N = f/D
So, minimum aperture this is simple: you just close the hole and have aperture of zero (f/∞).
But you can quite easily get below magical f/1 by the clever design. No need for diamond lenses, as Joanne C greatly explains. You can just grab a lot of light with front element as big as you want (D) to and squeeze it to the considered image (which relates to focal length).
In today's world you can meet this effect when using for example Metabones T Speed Booster 0.64 or 0.71 converter. It multiplies an focal length your lens by the number specified. So, if you get the beautiful Leica Noctilux f=50mm lens f/0.9 after using the Metabones 0.64 converter you get the effective f = 50mm * 0.64 = 32mm. Entrance pupil (as well as f) is proportional to sensor size d at given angle of view. So we move our lens+converter to a camera with d=35mm0.64 which gives ~23mm (sensor longer edge) - this appears to be micro fourd thirds system!. On this system our f gets back to 50mm, but D also gets multiplied by 0.64, so we have = f/(0.90.64) = f/0.576.
So were's the catch, you ask? Of course converter is not magic wand. It squeezes available light on smaller image circle, so you can use your Leica only on micro four thirds cameras. And added lens set affects image quality, but this is another story :)
This effect is also explained in cambridgeincolor lenses tutorial