Why don't cameras offer more than 3 colour channels?

It costs more to produce (producing more than one kind of anything costs more) and gives next to no (marketable) advantages over Bayer CFA.

(Or do they?)

They did. Several cameras including retailed ones had RGBW (RGB+White) RGBE (RGB+Emerald), CYGM (Cyan Yellow Green Magenta) or CYYM (Cyan Yellow Yellow Magenta) filters.

It seems to me that it would be very useful to have a camera with a greater spectral range and resolution, and so I'm wondering why cameras aren't available that capture more than three colour channels.

The number of channels is not directly related to spectral range.

Is there a huge technical challenge in making a sensor with four or more colour channels, or is the reason more to do with a lack of demand for such a feature?

The lack of demand is decisive factor.

Additionally CYYM/CYGM filters cause increased colour noise because they require arithmetic operations with big coefficients during conversion. The luminance resolution can be better though, at the cost of the colour noise.

Do multi-channel sensors exist as a research effort? Or am I simply wrong about how useful it would be?

You are wrong in that spectral range would be bigger with more channels, you are right in that fourth channel provides a number of interesting processing techniques for both colour and monotone.

Alternatively, if the does exist (or has in the past), which cameras have offered it, and what are its main uses?

Sony F828 and Nikon 5700 for example, they and few others are even available second-handed. They are common-use cameras.

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It is also interesting to know that spectral range is limited not only by the hot mirror present in most cameras but with the sensitivity of the photodiodes which make up the sensor. I do not know what type of photodiodes exactly is used in consumer cameras but here is an exemplary graph which shows the limitation of semiconductors:

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Regarding software which may be used to extract fourth channel: it is probably dcraw but it should be modified and recompiled to extract just one channel.

There is a 4x3 matrix for F828 in dcraw.c which makes use of the fourth channel. Here is an idea: { 7924,-1910,-777,-8226,15459,2998,-1517,2199,6818,-7242,11401‌​‌​,3481 } - this is the matrix in linear form, most probably every fourth value represents the Emerald. You turn it into this: { 0,0,0,8191,0,0,0,0,0,0,0,0 } (I do not know what number should be there instead of 8191, gust a guesswork), recompile and the output image gets the Emerald channel after demosaicing in the red channel (if I understand the sources correctly).