I power my Arduino Mini with 3 AA batteries on
RAW input. Is it possible to get the voltage on
RAW and therefore the expected longevity of the batteries?
No. Not directly. You can use voltage divider and measure the result with using internal reference. But it's another current consumption and you can't reduce it by (for example) entering one of the sleep modes.
And if you have
5V Arduino Mini variant, you can power it directly through
5V power pin. Otherwise you are losing about
1.5V on voltage regulator. On
3.3V variant it might be better, but it also depends on used regulator voltage drop and would be better to power it directly (if you don't have connected anything strictly with
3.3V power, and for
AVR based Arduinos it's not a problem)
Anyway, if you power it directly, you can measure the power source voltage by measuring internal bandgap voltage reference against power as reference. But, as far as I know, it can't be done by
analogRead, as it doesn't support other sources than analog pins.