A strong contender for favorite place status, the Zelve monastery complex is situated about 10 km out from Göreme on the Avanos road. Lacking the elaborate frescoes of Göreme and other sites there's still plenty here to see. The series of valleys can provide you with a couple of hours walking, climbing and crawling about and in addition to the marked highlights (the Fish and Grape churches) there are innumerable rooms and passages to look at.
Zelve was inhabited until quite recently but you can almost see the place crumbling before your very eyes. There's probably an element of risk involved in exploring too enthusiastically but a guide should be able to balance the thrill of stumbling through pitch black tunnels by torchlight with an element of safety.
It's probably a good idea to make the most of the place while there's still something to see. There seems little chance of a restoration scheme along the lines of that in place at Göreme and even if tourists were to stop visiting today natural erosion processes do their damage every winter.
Next to Zelve valley there is Pasabag, where you can see best examples of Fairy Chimneys with their so-called "hats", rock on them. In order to visit these sights there is an admission fee to be paid at the entrance of the valley.