What is the lowest acceptable storage unit temperature for video games? Also, at what temperature is there a noticeable effect on a video game’s battery save?
Storage temperature should be above around -40. at some temperature the metal will pull away from the things it’s attached to. Dunno about the battery operation: at some temperature freezing may damage it, but probably that would be well below zero.
Title edited to better indicate subject.
Celsius or Fahrenheit? 
What do you mean by “video games”? Consoles? Portable gaming devices?
Consumer electronics should be OK at -40 degrees for storage (non-operating). The major caveat is, it needs to be warmed up without condensation (dew) before use. Normally that means the item should be warmed up slowly, in very dry air. Ideally you’d want to put the device in a sealed bag (Ziploc or heat seal), along with desiccant, before exposing it to cold temperatures.
is a battery holding the memory? then the battery would be likely the limiting factor.
OP needs to rephrase their question. “Video games” are software that can be put on all manner of storage media.
Are we talking about old school cartridges?