Do you know what I like to do in my free time when I'm not making new circuits? Abusing for audio parts not meant for the purpose! Turns out the ubiquitous TL431 "precision programmable reference" (I have to thank for that how many I've salvaged so far) answers the "can it fuzz?" question with a surprisingly bold YES.
The idea came to me from the datasheet itself, showing a test circuit for voltage amplification. That looked surprisingly close to a common emitter amplifier, bias and all, which might seem surprising from the equivalent circuit. Of course the test circuit couldn't be used as it was because of the extraordinarily low impedances required to run the device at the minimum recommended current, the goal being precision and not at all fuzz. Given the similarity to a regular BJT stage though, coming up with a functioning one was surprisingly easy:
There's 4 versions for you to experiment with, given as starting points but also a working circuit already. In all versions I used DC-only feedback because of the very stable biasing, but without the drawback of AC feedback shredding my input impedance to pieces. Trying that resulted in a very weak output no matter what. The 2.2k load resistor is chosen because it's a good compromise for quiescent current, but 10k starts to be too much and it sounds sputtery.
The first version, probably my favorite, has no buffer and a somewhat low input impedance. The input cap is to taste in all versions, but here it works well to keep everything a bit tamer.
The second version is the same, just with higher input impedance and some value tweaks to keep the bias right. This one is more buzzy, unruly and gated, both because more signal gets through and also more bass (can probably be countered by a smaller input cap).
The third and fourth versions feature a DC-coupled buffer biased the same way as above, with two different input impedances. This is to show that this arrangement works and quite well.
Finally, I was surprised to find out that the usual way to adjust gain on a single stage without ruining your input impedance, which we might still call "emitter degeneration", works here too. You can hear in the demo below the effect of a 500R pot between anode and ground. AC coupling the pot in parallel with a resistor should work too.
Here's the demo of all versions, featuring also an earlier one with a standard AC-coupled emitter follower (with the warning that, unlike the DC-coupled version, that one is hard to bias):
This IC packs a lot of gain in a small TO92 (or smaller) package. Noise isn't the best, especially with the buffer, but the datasheet already puts our hopes down in that regard. Still, given how cheap and readily available these things are, I think it makes a good alternative as small and simple fuzz.