Mildly Infuriating
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I looked it up and the only direct 1.5V compatible AAA battery tech I found (source for the curious) seems to use lithium iron disulfide and are not rechargeable.
The rechargeables I found in AA/AAA format use lithium iron phosphate which outputs 3.2V so they must have internal circuitry (and hence losses) to make them output around 1.5V and thus be drop-in compatible with other AA/AAA (batteries in a standard format with non-standard voltages are a great way to get non-expert users to blow up their electronics devices). The internal electronics probably explains why they're so much more expensive than other rechargeables and (from what I found looking around) their capacity not be much more than NiMh (about 10-20% more).
nickel zinc are 1.6V, which could be fine for many things, but could still blow up a few things.
Well, battery voltage really changes a lot as the battery discharges (in that PDF I linked about non-rechargeable Lithium AA batteries there are some curves for Voltage based on charge which are pretty similar to other battery tech, except for the big drop at the very beginning) so battery devices are designed to be quite flexible with regards to the power input voltage and those which need very precise voltages include some form of DC-DC voltage conversion that makes sure the rest of the electronics gets a precise voltage.
Also circuits in Electronics are, as one of my EE teaches was fond to say, supposed to be designed with an "Oh, shit! margin", which is at least 10%, plus the actually components in the circuit also tend to have that or more margin in the power supply voltage they can handle (in fact in my experience most voltage sensitive components have a lot more than just 10%) per their specification, and they seldom blow up even if you go out of spec a bit (basically they're guaranteed to work by the manufacturer if you stay inside the spec, but most units of good quality components generally handle 10-20% out of spec conditions just fine, which is why for example you can overclock CPUs quite a bit before they stop working properly).
So providing 1.6V when it's supposed to be 1.5V should be fine unless that circuit is really badly designed.