What are other examples of characteristic numbers? Be warned, this may be a ridiculous question. 
I understand characteristic classes of principal $G$-bundles (and associated vector bundles) over a space $X$ arise from the classifying maps $f\colon X \to BG$ of those bundles by pulling back classes in $H^k(BG;R)$ along $f$, for some $k \in \mathbb{N}$ and coefficient ring $R$. For example, the Chern classes $c_k$ pull back from $H^{2k}(BU(n);\mathbb{Z})$ for big enough $n$, or from the colimit $H^*(BU;\mathbb{Z})$. 
If $X$ is an $n$-manifold, then taking a cup product, with total degree $n$, of characteristic classes of the tangent bundle $TX$, and then evaluating it against the fundamental class, one gets a characteristic number of the manifold. 
I know Pontrjagin numbers, Chern numbers, Stiefel–Whitney numbers, and the Euler characteristic arise in this way. But there are more characteristic classes I've seen named (although I couldn't tell you off the top of my head what $H^*(BG;R)$ they come from, and some I've never seen described that way; that would be nice to see too).
But it seems like there should be more characteristic numbers—there are after all more $G$ than $O(n)$ and $U(n)$ and more $R$ than $\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$. Or maybe I'm just embarrassing myself on the internet.
What are some other characteristic numbers?
 A: When you say "characteristic number of a manifold" instead of "characteristic class of a principal $G$-bundle" you're gliding over some important points. First, your manifold needs to be closed and oriented for it to have a fundamental class. Second, a priori the tangent bundle of an $n$-dimensional manifold only gives you a principal $O(n)$-bundle (up to homotopy), and in order to get a principal $G$-bundle out of this for $G$ some other interesting group you need to supply some extra data, usually a choice of reduction of the structure group from $O(n)$ to $G$, which also requires specifying a morphism $G \to O(n)$. In particular, you can't talk about the Chern numbers of a $2n$-manifold until you pick an almost complex structure on it, or equivalently a reduction of the structure group from $O(2n)$ to $U(n)$.
It's also worth pointing out that the classical cases of characteristic numbers you describe happen to be particularly important: they are complete cobordism invariants (for oriented and complex cobordism respectively depending on the characteristic number). 
So, having said that, here's an example that depends on a choice of principal $G$-bundle. Let $G$ be a finite group and let $\alpha$ be a class in $H^n(BG, U(1))$ where $U(1)$ has the discrete topology. If $M$ is a closed oriented $n$-manifold, then associated to the classifying map $f : M \to BG$ of any principal $G$-bundle on $M$ there is a characteristic number $f^{\ast}(\alpha) [M]$ called the Dijkgraaf-Witten invariant of $f$. This construction is important in producing a topological quantum field theory called Dijkgraaf-Witten theory; the next step is to integrate the characteristic numbers above over the space of all principal $G$-bundles on $M$ in a suitable sense. 
