There are various different types of limitation to take into consideration.
Effects for which the path of a ray is dependent on its wavelength
These are a class of effects for which spectral rendering is required, and a number of interesting examples have already been given in Benedikt Bitterli's answer. A simple example is a prism splitting white light into a spectrum, giving rainbow colours. Rays of different wavelength are refracted by different angles as they pass through the prism, resulting in the light striking the wall behind the prism being split into its constituent colours.
This means that in real life, shining monochromatic yellow light through a prism will result in yellow light coming out, but shining a mixture of red and green light that approximates yellow will result in separate red and green light emerging. When rendering using only 3 primary colours, white light will split into only those three colours, giving rainbow effects that look discontinuous, and monochromatic light that should not split at all will split into its approximating primary colour components. The splitting of white light can be improved by using a larger number of primary colours, but this will still give discontinuities close up, and the results for monochromatic light will still be split, albeit more narrowly. For accurate results a continuous spectrum must be sampled, with wavelengths distributed according to the light source being simulated.
Surface effects that cannot be captured in a single still image
Iridescence, for example, shows a different colour to each eye so that a still image will not look the same as the original object. There are many everyday examples that you might not notice at first. Many common birds have iridescent feathers even though they appear black or grey from a distance. Close up they are surprisingly colourful.
A renderer using only 3 primary colours will not be able to produce the spreading of light based on wavelength required for this effect. A spectral renderer can simulate the spreading correctly, but the full effect still cannot be captured in a single image. Even a 2d photograph cannot capture this correctly, whereas a 3d photograph of an iridescent object will give that shimmering effect as the photographs corresponding to left and right eyes will be coloured differently. This is a limitation of 2d images rather than the RGB colour space itself. However, even in a 3d image there will be colours in the iridescent object that are not displayed correctly, due to the inability of RGB to display monochromatic colours as described below.
Colours that the human eye can detect that cannot be displayed in RGB
RGB was historically device dependent and therefore unreliable between platforms. There are device independent perceptually uniform improvements such as the colour space Lab, but these are still trichromatic (having 3 components). It is not immediately obvious why three components is insufficient to display all the colours that can be perceived by a trichromatic eye, but this paper explains it well, and accessibly. From page 7:
For example, using a modern laser-display system with monochromatic primaries at 635 nm (red), 532 nm (green), and 447 nm (blue), lets see if we can simulate the perception of a monochromatic light at 580 nm (an orange color). Since the monochromatic orange stimulus excites the greenish and reddish cones, a contribution is required by both the green and red primaries, while no contribution is required from the blue primary. The problem is that the green primary also excites the bluish cones, making it impossible to exactly replicate the orange stimulus
The diagram of human eye cones sensitivities (also on page 7) shows how wide the overlap is and helps to visualise this explanation. I've included a similar graph from Wikipedia here: (click on the graph for the Wikipedia location)
In short, the overlap between the range of colours that can be picked up by each of the three different cones (colour sensors) of the human eye means that a monochromatic colour can be distinguished from an approximating mixture of primary colours, and therefore mixing primary colours can never accurately display all monochromatic colours.
This difference is not usually noticeable in everyday life as most of our surroundings emit or reflect light across a wide range of frequencies rather than single monochromatic colours. However, a notable exception is sodium lamps. If you live in a part of the world that uses these yellow-orange street lights, the light emitted is monochromatic and will look subtly different from a printed photograph or an image on a screen. The wavelength of sodium light happens to be the 580 nm from the example quoted above. If you don't live somewhere that has sodium street lights, you can see the same single wavelength light by sprinkling finely crushed table salt (sodium chloride) over a flame. The scintillating yellow points of light cannot be accurately captured on film or displayed on a screen. Whatever three primary colours you choose, there will always be a range of monochromatic colours that cannot be displayed.
Note that this limitation applies equally to mixing 3 primary colours of paint, using 3 photoreactive chemicals on a camera film, or taking a photograph with a digital camera with 3 different colour sensors, or a single sensor with 3 different primary colour filters. It isn't just a digital problem, and isn't just restricted to the RGB colour space. Even the improvements introduced by the Lab colour space and its variants cannot recover the missing colours.
Multiple diffuse reflections (colour bleeding)
If a brightly coloured matt surface is near a white matt surface, the white surface will show some of the colour of the other surface. This can be modeled reasonably well using purely red, green and blue components. The same combination of red, green and blue that gave the colour of the coloured surface can reflect off the white surface and show some of that colour again. However, this only works if the second surface is white. If the second surface is also coloured, then the colour bleeding will be inaccurate, in some cases drastically.
Imagine two surfaces that look a similar colour. One reflects a narrow range of wavelengths around yellow. The other reflects a wide range of wavelengths between red and green, and as a result also looks yellow. In real life, the light showing on one surface due to the other will not be symmetrical. Most of the light reaching the wide wavelength range surface from the other will be reflected again, as the narrow range of incoming wavelengths are all within the wider range. However, most of the light reaching the narrow wavelength range surface from the other will be outside the narrow range, and will not be reflected. In an RGB renderer, both surfaces will be modelled as a mixture of monochromatic red and monochromatic green, giving no difference in reflected light.
This is an extreme example where the difference will be instantly noticeable to the eye, but there will be at least a subtle difference in most images that include colour bleeding.
Materials that absorb one wavelength and emit another
joojaa's answer describes the absorption of ultraviolet light by snow, to be re-emitted as visible light. I hadn't heard of this happening with snow before (and frustratingly I've been unable to find any evidence to support it - although it would explain why snow is "whiter than white"). However, there is plenty of evidence of it happening with a wide range of other materials, some of which are added to clothes washing detergents and paper, to give extra bright whites. This allows the total visible light outgoing from a surface to be more than the total visible light received by that surface, which again is not modelled well using only RGB. If you want to read more about it, the term to search for is Fluorescence.
Eyes with more than 3 primary colours
There are animals that have more than 3 types of cones in their eyes, allowing them to perceive more than 3 primary colours. For example, many birds, insects and fish are tetrachromats, perceiving four primary colours. Some are even pentachromats, perceiving five. The range of colours that such creatures can see dwarfs the range displayable using only RGB. Far beyond them is the mantis shrimp, which is a dodecachromat, seeing colours based on 12 different cones. None of these animals would be satisfied by an RGB display.
But more seriously, even for images intended for human eyes, there are believed to be human tetrachromats who see in 4 primary colours, and possibly some who see as many as 5 or 6. At present, such people don't seem to be present in sufficient numbers to make displays with more than 3 primary colours commercially viable, but if in future it becomes easier to identify how many primary colours a person can see, this may become an attractive trait leading to it spreading throughout the population in future generations. So if you want your great grandchildren to appreciate your work you may need to make it compatible with a hexachromatic monitor...
Not really relevant to this question, but related: If you want to see colours that are not available in either the real world or RGB images, have a look at Chimerical Colours...