I was looking at pictures of a test fire of a TR-106 engine, and it appears to be fairly under-expanded.
What's the reason for this? Does this somehow increase efficiency with this specific engine?
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There's no good reason to run a production rocket engine underexpanded at ground level; it loses efficiency and will only become more underexpanded as it gains altitude.
If you compare the shape of that nozzle and the size of the combustion chamber to pretty much any other rocket engine, it's clear that this is a very short nozzle.
The curve of the expanding exhaust plume gives a pretty good idea of what the nozzle should actually look like -- 3-4 times the length of the existing nozzle stub.
Therefore, I believe this is is a test of the combustion chamber only. Since the TR-106 is ablatively cooled, there's no point in fabricating a disposable nozzle extension for early tests, and the more compact engine is easier to transport and install.