First, let's accrete onto a normal white dwarf. White dwarfs are white dwarfs because they have no hydrogen to burn. However, through accretion, the star can find itself with a substantial layer of hydrogen on its surface. We call this a nova. Now, let's fast forward a few years.
White dwarfs are white dwarfs because electron degeneracy holds up the star against gravity.
But electron degeneracy only works if the star is less than 1.4 solar masses. Normally, this is not a problem, but, over time, an accreting white dwarf may get enough material to put it over the Chandrasekhar limit. If this happens, crush, ka-boom, the white dwarf collapses to a neutron star in a supernova type event. Even a neutron star can be affected by accretion. Just like a person who strokes a basketball and makes it spin faster and faster, the accretion of matter onto a neutron star can make the star spin faster and faster.