I very much doubt there is anything you can do to gain privileges, unless "exploitme" itself has suid privileges [and there is no reason for such an application to be given suid].
In the example given by Jon Erickson in Linux Magazine, the application is given suid privileges, and after that, the stack overflow is used to make the application propagate those privileges to other functions in the system, for example by calling system(). But system in and of itself is not a function that can be exploited [at least as far as I can determine by reading the source of it].
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Mats
