Assuming a 3.5 digit DMM, the lowest digit would be 10mA, so you would expect it to read zero. Also some newer chipsets for low-end DMMs are not quite as good from an accuracy pov as the older ones based on the ICL7106 (and its many clones) and have more nonlinearity etc. but require fewer and cheaper/smaller external components.
If you put it (on the 20A range, briefly only) across a reasonably fresh 1.5V AA alkaline battery it should read a couple amperes, and maybe half that for an AAA cell.