One of the most widely used methods for solving equations is the classical Newton's method. While this method often works -- and is used in computers for computations ranging from square root to division -- sometimes, this method does not work. Usual textbook examples describe situations when Newton's method works for some initial values but not for others. A natural question that students often ask is whether there exist functions for which Newton's method never works -- unless, of course, the initial approximation is already the desired solution. In this paper, we provide simple examples of such functions.