Umm.... the closest galaxy to the Milky Way is the Canis Minor Dwarf Galaxy. The next closest is SagDEG (Sagittarius Dwarf Elliptical Galaxy). Only when NOT including dwarf galaxies is Andromeda the closest.
I second this. There are a number of galaxies closer than Andromeda - the two Magellanic Clouds, as well as those mentioned above. If you change the answer the dispute about the status of the Canis Major Dwarf Galaxy will come up, so perhaps change the question to specify 'non-dwarf' galaxy.