Technically there is no color space that can exceed the visible spectrum, because colors are made by the brain. What the camera collects is light of various wavelengths from long to short. The sensor just captures the photons from these various wavelengths and counts them up and says there were 2 million counts for the long wavelength, 3 million counts for the middle wavelength, and zero counts for the shortest wavelength, and if the human visual system experienced this mix of wavelengths it would create the sensation of something it calls yellow. The camera software attempts to emulate what the brain would produce by sending various signals to the monitor which has a matrix of red, green, and blue LEDs. It stimulates the right combination of LEDs to produce a sensation in the brain of "yellow." These signals are nothing like the original wavelengths of light coming from the scene - but they have the same effect on us as if we were at the scene - well almost. There is actually no "color" in nature.
So the purpose of color management is to create something from a monitor display, or from a film, or a print, that makes us believe we are seeing the wavelengths of light that were coming at us in the original scene. Ha-ha. So don't sweat it; the technology isn't available to do this right.
It is true that the cameras can detect wavelengths of light that we can't recreate in our brains - very long wavelengths and very short. In addition cameras can create combinations of signals that are beyond what the monitor can display because the RGB color system that is in most electronic display systems is very limited in what it can produce - no bright yellows and no deep reds for instance. Printers on the other hand can produce bright yellow, and a whole lot of other colors that the RGB system can't make.
So, do you want the final image to display colors you can't see on your monitor -If you have see "kodachrome" -like colors in the original scene, and these colors are important to you, you can set up your color management system to save those "color" conversions and move them to the printer, even though you won't be able to see them on the display monitor. The printer paper/ink selection can then print those "kodachrome" colors.
If you are shooting for the web, then you should start with sRGB color space and continue with it to the final JPEG for the web. Older web browsers don't color manage and anything other than sRGB leads to flat pictures. If your end viewer is looking at your pictures on a laptop or an LED display, don't worry about working in a color space bigger than the monitors can display - which is sRGB. (there are monitors with a bigger color space but they have more imaging issues that you would probably want to deal with. Barco sells them for $10,000 + for medical use.)
I use a big color space to work in and I do convert my captured data (either in camera or in the RAW converter) to at least "adobe 1998" color space, because I do a lot of PP on my images and I need "elbow room" to move colors around without them getting "scrunched" by a limited color space. In the end I have to convert to sRGB to show them on the web - though for my friends they get images profiled for something larger in case the end result is to be a print.
People that make a living producing heirloom quality prints spend a great deal of time and money dealing with color management issues, and I would defer to their teaching if you decide to head in that direction - much useful teaching is right here at TLL.