I'll kick this off with my solution because this is a fun one to see a bunch of different answers

a) A single object category whose morphisms are the Natural numbers, composition given by \$$x ; y = x * y \$$, and the identity morphism on the sole object is \$$0 \$$.

This obeys the associative law because \$$(x * y) * z = x * (y * z) \$$

This fails to obey the unit law as you can see by the example \$$0 * 1 \neq 1 \$$

b) A single object category whose morphisms are the natural numbers, composition given by \$$x ; y = |x - y| \$$, and the identity morphism on the sole object is \\$$0 \$$.

This obeys the unit law because \$$|0 - f| = f \$$ and \$$|f - 0| = f \$$ when \$$f \geq 0 \$$, which is true because \$$f \in \mathbb{N} \$$

This fails to obey the associative law as you can see by example \$$||1 - 2| - 3| = 2 \$$ and \$$| 1 - | 2 - 3 || = 0 \$$