Do you get any warning that the battery in a digital meter is on the way, out or does it just stop working.
On one of mine, the "-" turns into a small "batt", then the display goes wonky. A few months later. This usually only happens when I have forgotten to switch it off for a few days. Used properly, battery life in service is pretty much the same as its shelf life.
The best thing to do with any meter is to switch it to a resistance scale and tap the probes together before use. If it reads anything other than zero, something needs checking. From experience, immediately after doing that check is the time to switch to the intended range. Saves the hassle associated with offering the resistance measuring circuit to a voltage.
Each design has its own little differences, I wouldn't be surprised if there was at least one that could self check its own battery, but I suspect it would be at the upper end of the market.