Without making modifications to the components, a Nano Data Logger will last two or three days on small batteries (4 or 6 AA or AAA). I'm not sure larger batteries help too much because the voltage regulator might just waste the excess power, but properly chosen batteries will help some.
Disabling the bright LEDs on the Nano and shield will probably help quite a bit. Removing the voltage regulator on the Nano could also help (but I guess that limits the options for input power). I assume the shield also has a voltage regulator that uses power, but I guess you need that to use the screw terminals for input power.
If the processor and SD card cannot be put to sleep between logging events, the longevity while on batteries won't be very impressive. Ed Mallon suggested that the square wave output of the RTC could be used to wake things up, so it might be possible to sleep some components of the logger. Somebody who knows what they are doing could probably make the Nano Data Logger last more than a few days.
If the modifications to make the logger last longer require hard stuff like soldering, the advantages of the Nano Data Logger might be lost and using an Arduino Pro Mini could be an easier solution.