Not really. Classes/structs are just defining offsets into memory, so if you have a class within a class within a class, the compiler just adds up the offsets.
Performance comes into play once you have pointers (each pointer dereference is a memory read and potential L2 cache miss) or virtual functions (very bad, especially on older CPUs).
EDIT: One thing I should note though - if you're developing an application where performance is not absolutely crucial, focus on good class design rather than performance. While things like L2 cache misses make a big difference when you're writing something that needs to run at 60fps, it is of little relevance in a normal desktop application.