Any topology in which all finite sets are closed is $T_0$, and all of your topologies are such topologies.
The co-finite topology is the smallest such topology - the open sets are the sets $U$ with either $U=\emptyset$ or where $\mathbb R\setminus U$ is finite.
One class of topologies that does not contain the co-finite sets are the "order" topologies. The most common (and actually, useful) one is the left- and right-open topology:
$$\tau_{-}=\{(-\infty,a)\mid a\in\mathbb R\cup\{\pm\infty\}\}\\
\tau_{+}=\{(a,+\infty)\mid a\in\mathbb R\cup\{\pm\infty\}\}\\$$
These are actually useful - they are related to left- and right- continuity.
Give a topology $\tau$ on $X$, and an element $x\in\mathbb X$, there is a notion of localizing $\tau$ to $x$. An open set in $\tau_x$ is either any set not containing $x$ or any set which contains a $U\in \tau$ with $x\in U$.
Functions $X\to Y$ are "continuous at $x$" if $(X,\tau_x)\to Y$ is continuous. We can show easily that $\tau_x$ is a $T_0$ space even when $\tau$ was not.
Then it turns out that $\tau = \bigcap_{x\in X} \tau_x = \tau$. This roughly means that continuity on the entire space is the same as continuity at every point.
Now, an interesting question might be: What are the topologies ($T_0$ or otherwise) on $\mathbb R$ such that the standard binary operations $+,\times:\mathbb R\times\mathbb R\to\mathbb R$ are continuous. I don't have an answer to that question.