The infinite edgeless graph has trivial definable closure, since for any finite tuple $\bar a$, there are automorphisms fixing $\bar a$ and swapping any two vertices outside of $\bar a$.

But of course, the infinite edgeless graph is so not interesting to have drawn on the chalkboard. (I suppose you could address this by drawing the complete graph on infinitely many vertices, which also has trivial definable closure for the same reason.) But I would recommend another example, the countable random graph. For any finite tuple $\bar a$ and any vertex $b$ outside $\bar a$, there is an automorphism fixing $\bar a$ and moving $b$, and so the definable closure of $\bar a$ is itself. This situation would be a good excuse to discuss the random graph, if you have not done so already.