You just wind up with a tree. C has lines to C1, C2, C3. C1 has lines to C11 C12, C13, etc. A complex tree can quickly get too large to take in on a single diagram. That's where it's nice to distinguish between the model which contains everything and a diagram which shows some small part of the model for some specific purpose. For example, a diagram with C, C1, C2, C3 is perfectly valid for a particular level of abstraction. Another diagram with C1, C11, C12, C13 is good at another level.
Bookmark Scott Ambler's Agile Modeling site. I refer to it fairly often.
Diagrams of UML 2 Composition Scott pops in here now and then, too. He's pretty big on that "purpose" I mentioned above. I like to fill in "_______ reads this to learn ________ so they can ________" That will help you know what level of abstraction to put in, or whether to skip the diagram entirely.