RAW data linearly scaled down to 8-bits would result in a very flat, dull image lacking good contrast. For this reason RAW converters apply an S-curve to give much greater contrast in the mid-tones (unless you specifically ask for something else). Depending on where the end points and shoulders of the curve are placed you move the area which gets maximum contrast around (contrast is determined by the steepness of the curve at that point) whilst increasing/decreasing brightness etc.

It's important to separate dynamic range and bit depth. The two only ever depend on each other when the encoding is perfectly linear. In all other cases they are completely independent - if you choose an encoding which massively compresses the mid-tones then you can record a huge dynamic range using very few bits.

What I'm wondering is whether any programs provide a default mapping from HDR to 8-bit images one might call "compression," which would just transform each HDR pixel to an 8-bit pixel without clipping (but obviously losing dynamic detail across the spectrum).

A "high dynamic range image" in the purest sense is no different to a RAW image (aside from the Bayer pattern), they both have more dynamic range than can be displayed, just one has greater DR than the other.

When most people look at "HDR" images, they're really looking at low dynamic range images that have been generated by tonemapping the HDR data. This operation attempts to preserve some of the dynamic range by using the full range available in the LDR image to display each different area in the HDR image.

Imagine the scenario you mention earlier where you take a slice out of the data to go from HDR to LDR, now imagine that window is shifted left or right depending on where you are in the image. That's the fundamentals of tonemapping. There's a whole load of other stuff going on but that's what makes it possible to display an HDR image on a low dynamic range device (like a typical LCD monitor).

So, for example, can one ask for a "compressed" rendering of a RAW file in Lightroom? For each channel this would map 0 to 0 and 2^14 to 2^8, with other values mapped linearly in between.

You might be able to by selecting the minimum possible contrast settings, it's hard to say how much of a curve is being applied. But you certainly wouldn't want to do this in most cases as your image would look very dull.