Translation:My cell phone does not have any memory.
"My mobile does not have a memory" is currently marked as incorrect, general strengthening on web. I think 'any' or 'a' could be quite interchangeable here. 'a' is perhaps slightly slangy but I used to use it in that sense quite a bit when I was younger (not so much now that even fridges and toasters have 'memories').
I’m doing the beginning exercises for “abstract objects” and was presented with: «Το κινητό μου δεν έχει μνήμη.» Since I do not see the word “any“ in the Greek, I did not choose that tile. I was looking for the word ‘much’ which was not offered, figuring that all phones do have a memory chip present. Is there no equivalent for ‘any’ in Greek?