treewidth of a given graph
In the survey Treewidth, partial $k$-trees, and chordal graphs by Pinar Heggernes, we have the following lemma, whose proof is given as an exercise.


  Lemma. Let $(\{X_i \mid i \in I\}, T = (I, M))$ be a tree decomposition of $G = (V, E)$, and let $K \subseteq V$ be a clique in $G$.  Then there exists an $i \in I$ with $K \subseteq X_i$.


Prove this lemma, and conclude that, as Yuval answered,


  Corollary. For every graph $G$, $tw(G) \geq \omega(G) - 1$. 


Conclude that the treewidth is at least three.

The treewidth happens to be at most three as well, but that's a different exercise.