It is possible to do the GPS processing after the fact. You turn on the rx and listen for a few hundred milliseconds, then turn off. This is very low power. Later you push these to a server (a smartphone or the cloud) which computes the position. This is called snapshot processing.

So if you don't need the position in real time for navigation it is quite possible to track location over a week with a tiny battery. This could be time triggered and/or based on MEMs counters (which are also power friendly). Or you could calculate locally if the user explicitly asks.

https://ieeexplore.ieee.org/document/5708033 http://publica.fraunhofer.de/documents/N-464241.html