The traditional argument for not using rechargeables on low-power devices is that the self-discharge rate for the rechargeable battery is higher than the power consumption of the device itself. However, some NiMH batteries (e.g Eneloop, https://eneloop101.com/) seem to use a special chemistry that results in only a 30% discharge after 10 years. I use the Eneloop brand pretty much exclusively. I think the extra cost of the rechargeable batteries have more than paid for themselves by avoid the damage caused by leaking alkaline batteries. The cost of NiMH batteries are now low enough that they may be far cheaper than alkaline batteries for almost all applications.
Most chargers will just assume that you've discharged them fully between uses, and perform a "full" charge cycle when one is plugged in, maybe with a temperature sensor to stop if they get too hot. You can't really use resting voltage as a gauge, and it's tricky to find a good charger which can reliably detect the small and brief "dip" in voltage which occurs around the time that the batteries are fully charged.
They're great batteries, and I also prefer them for applications where the batteries get fully drained before being plugged in, but they can still be iffy for applications where you only discharge them briefly and want to keep them "topped up".
For those applications, these days you can buy 1.5V lithium cells in AA/AAA form factors with built-in microUSB charging ports. But those are lithium-based, so you have to be careful about thermal runaway.
Life is a balancing act full of compromises...
I didn't know about lithium AA/AAA rechargeable batteries. Interesting. I see that most are rated at 1200-1500mAh (AA) and 400-500mAh (AAA). These are only a little over 1/2 the capacity of the NiMH that I normally use. I guess the charging circuitry takes up room in the battery.
And yeah, the lithium AA/AAAs were news to me too. I learned about them when I got a headlamp that could either take 3 AAA batteries or a same-sized lithium battery with +/- contacts in the right places. I think the single-cell ones include a buck converter along with the charging circuit, which probably cuts down on the space they can devote to the battery itself.
By the way, this situation happens mostly on my motion-activated staircase LED lights which take 4 x AA. That thing is amazing because it works until 3 x AA are essentially dead and only 1 x AA is giving out power. For most of my other devices, the device stops working before the battery voltage becomes that low.
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Repeated overcharging can cause changing voltage curves that are (apparently) also referred to as memory effect, but NiMH cells are also prone to this.
After finding and reading an answer that agreed with my initial assessment I never continued to investigate why my friend would make such a claim, but now I guess it was related to this phenomenon. Guess I owe somebody an apology. Crazy how somethings seem so outlandish that I did not even ask myself if this was possible or what fundamental differences there are between batteries of different states of charge. Also it would have been quite easy to test this in retrospective... Note to myself: I should be more careful and curious in the future!
That alone tells me a dead battery of one composition will weigh some fraction less than a fully-charged one of the same composition, assuming all other things are truly equal (number of atoms in the containing can, etc.) The fractional difference might be very, very small (billionths of a gram difference?) but should still be present.
I found and submitted this article when looking for the physical basis of the effect afterwards.
I guess I was one of that day's lucky 10,000! https://xkcd.com/1053/