Yes, no annealing anywhere near the head. That can result in the gun blowing up in your hands.
Bryan Litz did some testing of cases annealed every reloading in the AMP machine alongside those that were not. Nine loads without annealing produced a 7.5 fps SD, while ten loads annealed every time got 6.9 fps SD. If you divide those numbers by the square root of the number of samples, you get the standard error, which is the expected standard deviation of the average value. These standard error values are 2.5 and 2.3, respectively, meaning the SD is expected to be more than ± that much about a third of the time in future nine-shot samples. So you would get the two trading places much of the time and you don't know if the test-firing was one of those times or not. So I see this as no statistically significant difference.
I believe the reason for Litz's result is simple. While brass hardness lowers the percent of elongation the brass can withstand before breaking and it lowers malleability by raising the elastic limit and yield point (makes it more springy), it does not change the property called the modulus of elasticity, which reflects how much force it takes to stretch the brass a certain amount within its elastic range. Therefore,
if the brass is resized so seating the bullet stretches the neck the same amount each time, the amount of bullet pull and its resulting start pressure will stay the same, regardless of hardness.
There are several complications to the above. For one, brass annealing is a complex business. It goes through three stages, and in order of the temperature elevation needed to achieve them in brass, they are:
Recovery
Recrystallization
Grain growth
Recovery is all that is needed to stop case neck cracking. You don't need recrystallization and don't want grain growth, as the latter weakens the brass.
Recovery happens at the atomic level and softens brass by lowering the concentration of displaced atoms that working the brass made slip from their original crystal lattice locations (aka, dislocations). Dislocations exist under stress because the atoms want to go back where they belong. For that reason, recovery of atomic locations (also called partial annealing) accomplishes stress relief in the metal. Recovery can be measured non-destructively either as a lowering of electrical resistance in a given length and cross-section of the brass or by X-ray diffraction. Destructively, it can be measured as an increase in the percent the brass may be stretched before it splits or by sectioning the brass and doing a hardness test. A surface hardness test on a case neck is also possible that is not substantially destructive, but the radius of curvature and surface oxides make that trickier, so it takes some expertise to compensate for the errors those factors introduce. Because recovery happens at the atomic level, it produces no changes large enough to be seen in a metallograph. It happens at all temperatures, including room temperature (but is terribly slow at room temperature; think on the order of decades to achieve the observable effect).
The higher the displacement concentration, the lower the combination of temperature and time at which rapid recovery will initiate. That last point has a technical consequence. When brass approaches 100% work hardening and is getting very brittle and ready to split, recovery can be initiated and taken far enough to prevent neck splitting at time and temperature combinations that are well below temperatures that initiate recrystallization. However, if the brass is work-hardened to a lesser degree, it can show no response to the same time and temperature that stress-relieves a very hard neck pretty completely. So you can anneal every time you reload, but you may be having no measurable effect on the metal until it passes a certain hardness level which may not occur for several reloadings.
