Maybe the simplest category without a terminal object is \\( \textbf{2} := [ 1 \, 2 ] \\) the discrete category of two objects.

Does \\( \textbf{0} \\) qualify? It has no morphisms (to any object) but it has no objects from which to draw a terminal-object so it has no terminal-object.

The category \\( \textbf{1} \\) has a terminal object but the category with one object and one morphism in addition to the identity-morphism does not have a terminal-object [uniqueness is violated].