A neutron star has a mass that is up to twice that of the sun but a radius of only a dozen kilometers. Therefore, it has an enormous density, thousands of billions of times that of the densest element on Earth. An important property of neutron stars, distinguishing them from normal stars, is that their mass cannot grow without bound.
If a nonrotating star increases its mass, its density will also increase. Normally this will lead to a new equilibrium and the star can live stably in this state for thousands of years.
However, this process cannot repeat indefinitely and the accreting star will reach a mass above which no physical pressure will prevent it from collapsing to a black hole. The critical mass when this happens is called the "maximum mass" and represents an upper limit to the mass that a nonrotating neutron star can be.