Most often called a "wash" by masons, the chimney top is often a source of water leaks that is initially attributed to chimney and roof flashing.

The chimney flashing in the photo below looks like it could be the source of roof leaks. But the top of the chimney is also in need of repair and is likely  contributing the the roof leaking.

In the past most chimneys were topped off with the same mortar used to set the stones or bricks of the chimney. The masons would run their chimney flues, or clay pipes, above the top edge of the brick to prevent water, either from rain or snow buildup, from running down the inside of the chimney flue. They would then lay a thick layer of mortar on the top of the chimney from the flue, a few inches above the level of the chimney bricks, and taper it down to the outer edge of the chimney. This made a nice sloped top for the water to run off. Unfortunately, many times the mortar was not very strong and would not hold up to many years of weathering. Cracks start to form around the flues radiating out to the chimney edges. The picture below is typical of most chimneys. Once the chimney mortar starts to crack, water seeps in and under the top and the constant freeze / thaw cycles expand the cracks and lift the mortar cap. The cracks get bigger and allow more leaks into the chimney interior damaging the lower bricks and mortar as seen in the lower photograph.