Are stem height extenders safe to use?

This is a very good question, and one that a lot of people get wrong or approach thoughtlessly.

The answer is no, it's not significantly different in terms of safety whether your stem got where it is using an extender versus an uncut steerer.

The spacer stack restrictions on carbon and aluminum steerers are concerned with limiting the amount of leverage applied to the steerer right above the upper bearing contact. That's their entire purpose. You can't use an extender to cheat the restriction, because the amount of leverage generated will be virtually the same.

However, another side to this is that manufacturer spacer stack restrictions are (or at least should be) specified with the worst case scenario in mind in terms of rider weight, stem length/rise, and rough usage. So in the scenario of, say, a light rider with a short stem who needs to get up higher, the argument can be made that the restriction can be cheated some amount on that basis. Unfortunately there is no magic rule that will tell you by how much, because the manufacturer is always the only one that has actual data on material, heat treatment, etc. What it boils down to though is that there are many light riders on aluminum-steerered road bikes with stem extenders that get away with it just fine, although it is an at-your-own-risk thing. Extender manufacturers tend to be weirdly blase about this issue.

Steerer extenders should never go on carbon steerers because the stakes of the steerer failing without warning are too great; even in the case of a carbon steerer that was cut too short, they should be avoided because of the extra stress their pinch bolts put on the steerer. Carbon steerers should be handled exactly as their manufacturers specify, and none of them would condone an extender. Manufacturer instructions for extenders do also tend to say they should never go on carbon.

With normal wall thickness (1.5-1.6mm) chromoly steerers (as opposed to the more exotic steel ones a few companies have put out over the years, and those are generally pretty tough as well) you can make it as tall as you want and it will be fine.

I think the important question is stem length and not stem height. The largest component of the maximum force vector applied to the handlebars is force coming down from above. Minimal force is forward or side to side onto your handlebars while riding. The stem length creates a force mechanical advantage onto the headset and headtube, and the stem height does not. A very short 35mm (1.4") stem 8" above the headtube would likely transmit far less force onto the headset and headtube than a 120mm (4.7") long stem resting flush on the headtube, roughly three times less force.