Electronic – the outer metal casing of an alkaline battery made of

If we talk about small AA, AAA and coin batteries (both 1.5V and lithium 3V), then the simplest estimation can be made using a voltmeter. The voltage goes down (declines) over time and usage.

Note that each particular device needs a different minimum voltage to operate normally, so for example a battery from one device which seems dead can be put somewhere else and operate normally (at least for a little while). For example my digital weigh is quite demanding - it ceases operation already when the voltage drops to 2.88 V, and those lithium batteries are still well usable in bicycle computer or somewhere else.

Update: Note that alkaline batteries have got some kind of self-healing effect. When they are not used for a while, their voltage goes a bit higher. This can mislead your voltage measurements. So you should put them back to some load for a few seconds and then you can see their real health. Also note that totally dead batteries don't need this; they show voltage near to zero even without any load.

In the data sheet of the MAX6775, the device itself draws 3-7 uA, plus your voltage divider for the LBI input. The input has very low (5 nA) LBI bias current, so you could get away with some very high-valued resistors for your divider (for reference, if you have 5 nA bias, you would still be ~1% accurate with a divider that draws 500 nA, or 0.5uA, adding little to the chip itself.) And in the 5 uA range for the monitor circuit, the time it would take for your circuit to discharge the batteries would be on the same order as the self-discharge/shelf-life of the batteries themselves (using AA's for example, and assuming we aren't talking coin cells, but you didn't specify).

If you need to get it lower, you could use a high-side MOSFET to disconnect the battery monitor from the batteries, but then again, your micro is likely going to be drawing several microamps of its own in sleep at the very least.