The odometer display will draw about as much current as an LCD clock, so it will flatten the battery.......
.......after a few centuries.
Let's say a digital clock runs for a year on a AA dry cell with a capacity of 1000mAH. that's 1.5 Volts, one Ampere Hour, or 1.5 Watt Hours.
A car battery has a capacity of about 60 Amp Hours, at 12 Volts, that's 720 Watt Hours.
So ignoring any other factors, a car battery has the capacity to power a digital clock continuously for something like 500 years before it would need recharging.