I've heard that there are some things one cannot do as a computer programmer, but I don't know what they are. One thing that occurred to me recently was: wouldn't it be nice to have a class that could make a copy of the source of the program it runs, modify that program and add a method to the class that it is, and then run the copy of the program and terminate itself. Is it possible for code to write code?
Start by looking at quines, then at Macro-Assemblers and then lex & yacc, and flex & bison. Then consider self-modifying code.
Here's a quine (formatted, use the output as the new input):
#include<stdio.h>
main()
{
char *a = "main(){char *a = %c%s%c; int b = '%c'; printf(a,b,a,b,b);}";
int b = '"';
printf(a,b,a,b,b);
}
Now if you're just looking for things programmers can't do look for the opposite of np-complete.