Motivated by frequency-assignment problems in cellular networks, we study complete conflict-free colorings of graphs. A coloring of a graph $G$ is conflict-free if every vertex has a uniquely colored neighbor in its open neighborhood. The minimum number of colors required for such a coloring is the conflict-free chromatic number, denoted $\chi_{CF}(G)$. We determine the exact values of $\chi_{CF}(G)$ for several graph families such as paths, cycles, trees, complete graphs, and windmill graphs and provide conjectures on circulant and grid graphs.