How can you measure a wide range of temperatures without exotic materials? High-temperature thermocouples and thermistors typically use things like platinum and iridium because they don't oxidize in air.
A cartridge heater, as I understand it, consists of a resistive heating element coiled up, packed into a solid insulator made of something like MgO or BeO, combining very high resistivity and high dielectric strength with high thermal conductivity; all packed into a metal can to exclude oxygen and avoid abrasion of the relatively weak insulator (very weak in the case of MgO).
You can get fairly fine (≈100μm) copper wire by, among other things, untwisting Ethernet cables or some power cables. Annealed copper's temperature coefficient of resistivity α is about 0.0039, comparable to platinum's. You could maybe stick such a coil of fine copper wire inside some kind of insulator, such as quartz sand, and do a four-wire resistance measurement on it, using more copper wires twisted onto it.
To be concrete, 200 mm of 100-μm-diameter annealed copper wire should be about 439 mΩ at room temperature, about 610 mΩ at 120°, and about 2.15 Ω at 1020°, close to copper's melting point of 1085° (though probably α diverges a bit by then). If you were to dump a 1-amp pulse through it through two of the wires for 100 μs, then it should develop a voltage on the order of 400–2000 mV across the other two wires; 100 μs is plenty of time to measure the voltage and current waveforms even with a 44.1 ksps ADC, much less a 1 Msps ADC.
What error should we expect? Let's suppose we can calibrate out the thickness of the thermistor, copper's nonlinear temperature-resistance curve, and the impurities in the particular copper wire we're using.
An input impedance of 1 MΩ on whatever voltage measurement thing you have connected to the other two wires would give you a parasitic offset current on the order of 1 μA, causing a relative error of 10⁻⁶ or so on the temperature measurement. You can get op-amps with much lower offset current than that; they would provide a more precise measurement. For example, TI's datasheet for the LM741 — not a spectacularly low-offset-current op-amp — says its offset current at 25° is typically 20 nA, worst-case 200 nA, and the whole bias current is typically 80 nA, worst-case 500 nA.
The current you measure on the current wires outside the device would differ from the current through the "thermistor" only due to parasitic capacitances around the thermistor; if these were, say, 10 pF, then the current error would be around 10 pA, a relative error of 10⁻¹¹. For this error to rise to 10⁻⁶ you would need 1 μF, a totally unreasonable amount of parasitic capacitance.
Parasitic inductance is larger, but it's less of a problem. Suppose you have 100 nH of parasitic inductance in each pair of leads, which you can reduce by keeping them as closely antiparallel as possible, and another 100 nH in the "thermistor" itself (which you could reduce by alternating winding directions, FWIW). This would be very large; axial-lead TVS diodes are around 10 nH while surface-mount devices are closer to 4 nH. And suppose the current ramps up to 1 A in 10 μs. This produces 10 mV of voltage drop on the current leads and 10 mV of inductive voltage in the thermistor superimposed on the actual desired voltage. The first doesn't matter; the second, without any further measures, would produce a temperature error of some +6° during that time. Since the current through the voltage-sensing wires is sub-microamp, the error from their inductance is a million times smaller.
The reason this is less of a problem is, first, once the current finishes ringing, for the rest of the 100-μs pulse, the inductance introduces no error at all; second, you're probably applying more voltage than that, so the ramp time is even faster; third, if you take two or more samples of the voltage and current during the pulse, it's easy to decompose the results into an inductive component and a resistive component.
Heat is the biggest error. 1 amp through 439 mΩ is 439 mW, which is a lot. In 100 μs, it's only 43.9 μJ, which is not a lot. But this is a small wire; at copper's density of 8.96 g/cc, it's only 14 mg of wire. Still, at copper's molar heat capacity of 24.440 J/mol/K, which divided by its atomic weight of 63.546(3) g/mol gives 0.38460 J/g/K, that's a temperature rise of 8.2 millikelvins, 14 μV. Crudely, this would introduce an error of about 3 × 10⁻⁵ in the measured voltage, about 30× larger than the other sources of error discussed above.
However, this heating error is zero at the beginning of the measurement process, and on short time scales it is proportional to the integral of squared current so far. (Over slightly longer time scales the heat will start to leak away.) So I think it's probably relatively straightforward to cancel this source of error, at least down to the same 10⁻⁶ level as the offset current. If your ADC is fast you might be able to just use a 3-μs current pulse.
The whole process of generating heat and conducting it away from the wire into the rest of the device can be fairly closely approximated as a linear time-invariant process, so we can estimate its impulse response function, then cancel it, probably down to a 10⁻⁸ level, particularly if you stimulate it with random current pulses rather than regularly spaced ones.
It might be hard to get such a low error in the measurement of voltages and currents, due to things like drift, noise, and the temperature coefficients of the measurement device. Still, it's achievable, and even an 0.1% error in voltage would be an error of 1 K, which is enough for most of my purposes.
All of this is really quite astonishing, and it makes me wonder if I am overlooking some kind of enormous source of error.
It's important that thermal expansion and contraction not change the resistance of the current path, for example by tightening and loosening connections. For the voltage-measurement path, this is less important. So probably the right way to make the device is by tying and/or soldering some voltage-measurement wires onto the relevant part of the current-measurement wire.
Carbon, being a semiconductor, has a temperature coefficient about an order of magnitude higher, and it can also handle higher temperatures than the 1085° of copper. You can get carbon-composition and carbon-film resistors already in sealed ceramic "cases", though generally those aren't built to handle more than about 300°. (The paint on traditional axial-lead packages is organic and burns. Maybe surface-mount packages are better.)
Steel wire and stainless steel wire have temperature coefficients similar to copper's, but can withstand higher temperatures without melting, typically up to 1400° or more, though they must be protected from oxygen at these temperatures. They are better than copper for use in hot-wire cutters (at temperatures low enough that they don't burn) because they are stronger. They're typically thicker, though; stranded picture-hanging wire from an art store has the thinnest stainless-steel wire commonly encountered.
Tungsten wire is also similar, and comes much thinner in conveniently packaged US$1 quartz-halogen lightbulbs to protect it from oxygen, though perhaps not all of those envelopes will themselves withstand temperatures over 1000°. (And these lightbulbs can do double duty as heating elements.) However, they don't normally have four wires. I have such a 28W 220V lightbulb here; at room temperature (20°?), this untrustworthy multimeter, which measures a short circuit as 10Ω, measures it as 147.3–148.3Ω, so maybe it's actually 138Ω or so. In theory tungsten's α should be around .0045, according to the Wikipedia table above, so at 100° it ought to be about 188Ω and measure about 198Ω. I boiled it for 20 minutes on the stove, miraculously without breaking it, and it measured 185–186Ω. But the multimeter had gone to measuring a short circuit as 3Ω, so maybe that's really about 183Ω. Not the most spectacular agreement with theory but it's within measurement error. Next day short-circuit reading is 0.3Ω and bulb reading is 141Ω. (To work at 28W at 220V it would need to increase to 1700Ω; if it's at 1100° then its average α over that interval would need to triple, which is totally plausible. Some 70W-rated halogens I bought measure 51Ω so it's even more plausible.)
Silicon carbide, commonly used for heating elements and abrasives, is also a semiconductor, and can withstand even higher temperatures (decomposing at 2830°), protecting itself against air up to about 1600° by oxidizing the surface to silica. I don't know what its temperature coefficient of resistance is, but I imagine that it's negative and larger than copper's.
When using the same element to sense temperature and raise it, you can either measure the average temperature to which the element is heated (though not the peak, which is what you usually really want) by measuring the voltage and current while it's on, or the temperature of its surroundings by turning it off and letting it equilibrate.
This turns out to be what's standardly called a "resistance temperature detector" or "resistance thermometer", which is commonly used for high-precision temperature measurements, usually with a precision more like 60 millikelvin than 1 millikelvin. Copper is indeed a material commonly used for them, though usually not above 150° because of oxidation; other common materials are nickel, despite its poorer coefficient, and platinum; the most common configuration is Pt100: platinum with 100Ω at 0°. This higher resistance reduces the influence of lead-wire resistance.
I don't think copper is going to be able to reduce silica or magnesia, but if that turns out to be wrong, as a last resort you could pack it in copper oxide.
Wikipedia explains that above 660° your sensing filament tends to get contaminated with metal from the can it's in, so people usually use thermistors at higher temperatures despite their poorer precision and repeatability, typically having errors on the order of 2°. Because RTDs typically have microscopically thin wires flapping around, like lightbulbs, they tend to be less durable than thermocouples. The alternative to flapping around is to use a "wire-wound RTD" where the wire is wrapped around an insulating winding core, but the winding core generally has a different thermal coefficient of expansion from the metal, which causes strain in the metal, which introduces at least nonlinearity and maybe nonrepeatability.
The problem with picking up metal from the can seems like it could be reduced by using a can made of fused quartz or sapphire or something, or using carborundum and periodically burning off any metal that has deposited.
Many thanks to Greg Sittler for the very informative discussion!