Let the category $\mathcal{V}$ consist of elements of $\mathbb{R}^+$ ($[0,\infty]$) and give it a monoidal structure with $+$ as the tensor product and $0$ the identity element. Let morphisms in $\mathcal{V}$ be $\operatorname{Hom(a,b)}=\cases{*\quad a\le b\\ 0\quad a< b}$ for $a,b\in \mathbb{R^+}$. Now let $\mathcal{C}$ be enriched in $\mathcal{V}$. You can show that the morphisms of $\mathcal{C}$ are $Hom(a,b)\in [0,\infty]$ and that the notions of the triangle inequality and the identity are generalized in $\mathcal{C}$ as $$ \operatorname{Hom}(a,b)+\operatorname{Hom}(b,c)\ge \operatorname{Hom}(a,c)\quad\text{ and }\quad 0\ge \operatorname{Hom}(a,a) $$
My question is, since $\mathcal{C}$ is a categorical generalization of $\mathcal{V}$ then $\mathcal{C}$ must encapsulate other categories similar to $\mathcal{V}$. What are some examples of categories similar to $\mathcal{V}$ that when enriched give $\mathcal{C}$?