Is it only red, green and blue that can make up any color through additive mixture? I'm reading about color vision and have some trouble understanding the motivation for why the trichromatic theory was suggested in the first place. The book I'm reading ("Psychgology: The science of mind and behavior") states:

At the beginning of the nineteenth century it was discovered that any colour in the visible spectrum could be produced by some combination of the wavelenghts that corresponds to the colours blue, green and red in what is known as additive color mixture.

From the explanation in the book, it seems like this somehow should be a reason for also supposing that the human retina was composed of cones sensitive to the colors green, red and blue respectively. I guess this would be a valid argument if it was only red, green and blue that, through additive mixture, could make up any color.
Is it only red, green and blue that, through additive mixture, can make up any color?
 A: The human vision has 3 types of cones. (that is why all perception-based color spaces are 3 dimensional: LAB, XYZ, HSV ). Each cone type has a different sensitivity curve in the color spectrum (think of them as color filters). It gets complicated because these curves overlap: there isn't a single wavelength of light that triggers just one cone type.
So, in additive color synthesis, it would be nice if there would exist 3 colors that trigger the cones independently. By mixing light of these 3 base colors, you could create any color perception. But such a color set does not exist. RGB does a pretty good job of covering a large part of the color gamut, but not all (RGB fails at saturated cyan and yellow, for example).
In the XYZ space, you can see this quite visually: the total gamut has the shape of a horse footprint. An RGB color device can produce a triangle within. See the illustration in this question:
https://stackoverflow.com/questions/2455503/cie-xyz-colorspace-do-i-have-rgba-or-xyza
The curved edge of this gamut representation are the monochromatic colors. These are the toughest to reproduce in additive color synthesis: you can only produce a monochromatic color by producing its exact peak spectrum. 
You could consider adding a 4th and 5th primary color to an additive color reproduction system (e.g, an RGBCY monitor). That would be able to reproduce more colors, but apparently that has not been economical to this date. 
A: Your retina has cones that, roughly speaking, are excited by red, green, and blue (but spread out).
So any color you see, you see in terms of those three.
Any single visible wavelength of light excites these to some degree.
So, for example, a beam of strictly yellow monochromatic light excites the red and green cones, but not the blue so much, because yellow falls between red and green in the rainbow.
If two beams of light, one red and one green, were mixed in your eye, the result would look yellow.
Since you can't actually see yellow, only red and green, your eye can't tell the difference between pure yellow and two colors together, red and green.
But if you put those color beams through a prism, you could easily see the difference.
A pure yellow beam would be refracted in a single peak,
but the beam consisting of red and green together would separate into two distinct peaks, one red and one green.
In fact I'm pretty sure there is a color your eye can see that doesn't exist as a pure color in the spectrum.
If red and blue are mixed, you see kind of a light redish purple called "Magenta".
I don't think there is any pure wavelength of light in the rainbow that can excite both red and blue cones and leave out green.
So it's all just an evolutionary accident of how eyes got made, that they try to distinguish many colors by using a poverty of sensors.
A: Not any triplet of colors can make "all" the colors a human can see, and any triplet of colors can produce a variety of it.
It's just that RGB is the most efficient of all
A: Color printing and representation is tricky. CMYK covers approximately the same Gamut as RGB, but to get really good color printing printers from big manufacturers such as Canon, Brother, etc., have these days introduced 10 or 12 ink color printing. Each ink is chosen carefully to stretch the range of colors that can be printed to be as close as possible to the range of colors that can be distinguished by the eye.
Photographers often take their study of the issues as far or farther than many scientists. Google for "color gamut" gets all sorts of articles. For example, Luminous Landscape is quite a well-known photo site that includes many pages that include the word gamut, many of which are quite technical, but you could look at the fairly general page http://www.luminous-landscape.com/reviews/accessories/fancy_graphics.shtml to get an idea of the sort of extremes that photographers who want to print their photographs well will go to.
Of course, the artistry of printing well starts to be somewhat outside Physics, although a lot of it is knowing your materials very well.
EDIT: As the comment from Kris van Bael says, indeed the Question explicitly mentions "additive color". O dear. He's also right that the Gamuts of RGB and CMYK are different. Indeed the Wikipedia page I cite above has the following, "pure red can be expressed in the RGB color space, it cannot be expressed in the CMYK color space; pure red is out of gamut in the CMYK color space." It's precisely because of such gamut differences that printers have moved to larger number of inks. I've left this Answer here as a cautionary tale (for me if not for anyone else) rather than deleting it. O dear.
