Colour gamuts are complicated as colour is a largely neurological thing (not the simple physics of wave lengths the "magenta does not exist" crowd would like you to believe).
If you imagine a typical colour wheel (YRMBCG, the middle being a neutral grey), you can take three points on that wheel (we call that a triad), and from the colours you chose, you can "mix" any of the colours inside that triangle, and none on the outside of it. So if you pick a pure saturated CMY, you can mix all the things inside the triangle you would draw between those colours. The red, green and blue will not be the purest saturated possible colours, but as it happens, we almost never actually need purely saturated colours in printing or painting. When we do need them, we can just reach for a pigment different pigment. So fancy printers will let you do exactly that, you can keep to CMYK, but add in additional inks (this is one of the main uses for pantone colour names).
Our brain also does lots of interesting stuff with colour in close proximity. An extreme example of that is a Zorn palette in painting: vermilion (red), ivory black (a cool, very faintly blue leaning black), white and yellow ochre. There are lots of colours you can't reach with this palette, but you can get the brain to interpret some colours as leaning a bit more blue, or even green, by what colours they are next to.
If you take the colour wheel mentioned about, there are a couple of extra aspects that are worth visualising. Image that wheel in the middle, then take a point and stretch it up, so you get a cone, and imagine that is pure white, then stretch a second point down int he opposite direction, imagine that is pure black. That's the 3d "shape" of the typical colour gamut we think of, but we can get more accurate. For instance, not all hues (colours), are equally saturated at what we perceive as maximum saturation. The bluest blue, is darker, than the yellowest yellow. So if we imagine that that middle circle isn't actually a circle, but each hue is proportionally as far out as the brighest it can get (hard to describe), we end up with this wobbly shape. If you do the double cone thing but with that shape instead, that's more or less all the colours and tones we can see).
Then it gets even weirder because we can perceive different granularity of tones in different hues (we can distinguish a much larger range of greens than any other hue).
That before you even get to colour blindness or even tetrachromacy (some women can see more colours than the rest of us).