What are some examples of emergent abilities? By emergent ability I mean this: suppose that competently performing ability Z is contingent on, and only on, having skill X and skill Y developed to a certain level (that is, ability Z is a non-linear combination of ability X and Y that is distinct in type from those abilities). 

You would see a person developing, slowly, their abilities X and Y, and then, when they have developed them sufficiently, one would suddenly observe ability Z. 

I am interested in this question because it might seem like ability Z is a neurally innate ability that is suddenly triggered by some experience, when in fact this ability is just a combination of other abilities.

My question is perhaps two-fold: Do these sorts of abilities exist, and if so, what are some examples?
Many tasks can be decomposed into subtasks. And in a similar way the skills required to perform a task can be decomposed into component skills. For more detailed discussion, check out discussion of task analysis.

A few examples:


Driving a manual car requires an additional component skill (using the clutch and gearshift) over and above all the skills required to drive an automatic car (knowledge and application of road rules, speed control, vehicle control, etc.).
Writing an email requires a wide range of skills including basic skills in using computers, skills in communication and writing, and so on.


Much of education is based on the idea of layering new skills on top of those already established. For example, reading, writing, numeracy, and computing skills are fundamental to a lot of what is taught at university. 

Another perspective on your question is to think about transfer and training.


Transfer: Skills in many tasks involve transferring what we already know and then developing some task specific adaptations. For example, if you were to write a report in industry after starting a new job, you would build on all your previous training in report writing, but there would also be some domain specific learning related to the new job. In this sense, the previous acquired skills assist performance, but domain-specific learning is required.
Training: Training, practice, and experience also tend to have a certain structure. Development of skill in an overall task tends to lead to development of component skills. For example, when learning to drive a car, an individual will tend to get practice in all the component skills. The nature of experience itself will encourage development of steering, road navigation, and gear changes. Training can also be designed to deliberately compartmentalise skill development (e.g., see the part-whole training discussions [Naylor et al (1963)]).


Relating all this to your specific question, defining component skills and emergent abilities as discrete entities is difficult. Transfer is often relative, and each task in a new context or with different details can alter the dynamics of transfer and performance.

References


Naylor, J. C., &amp; Briggs, G. E. (1963). Effects of task complexity and task organization on the relative efficiency of part and whole training methods. Journal of Experimental Psychology, 65(3), 217.