How long will we continue to be able to see total eclipses of the sun?

The orbit of the moon is not stable. Because of tidal friction, the orbit of the Moon is steadily growing larger, so that the angular size of the moon from the Earth is shrinking. The moon's orbit is increasing by about 3.8 cm (1.5 inches) per year. When the moon's mean distance from Earth has increased an additional 14,600 miles, it will be too far away to completely cover the sun. This is true even at perigee when its disk will be smaller than the sun's disk even when the sun is farthest from Earth at aphelion. At the current rate that the moon's orbit is increasing, it will take over 600 million years for the last total solar eclipse to occur. A complicating factor is that the size of the sun itself will grow slightly during this time as it evolves as a star, which will act to make the time of "no more total eclipses" a bit sooner than 600 million years.