First let me say something that I don't completely understand, since I do not know enough physics. If I say anything wrong, someone please tell me:
For the spinning particle, there is a sigma-model, which is a type of quantum field theory, which describes how a spinning particle can propagate. The input data for this sigma model is an oriented pseudo-Riemannian manifold $X$ equipped with a line bundle with connection. The condition for "quantum anomaly cancellation", can be shown to be that the classifying map $X \to BSO(n)$ corresponding to the tangent bundle has a lift through the map $BSpin(n) \to BSO(n)$. Such a lift is called spin-structure.
The sigma-model for the spinning string, starts similarly, but the role of the line bundle is replaced by that of a bundle-gerbe (that is a gerbe with band U(1)). - I believe what is going on here is that a line bundle is the same data as a principal $U(1)$-bundle, and a bundle-gerbe is the same data as a principal bundle for the $2$-group $[U(1)\to 1]$. Anyhow, for this new quantum-field theory, the condition for "quantum anomaly cancellation" is that the classifying map $X \to BSO(n)$ has a lift through $BString(n) \to BSpin(n) \to BSO(n)$. In fact, $String(n)$ does not exist as a Lie group, but it does exist as a (weak) group object in differentiable stacks, which are in particular sheaves (over the category of manifolds) in homotopy 1-types.
Apparently this can be taken even further, and one can talk about a sigma-model for the so-called $5$-brane, and the condition "quantum anomaly cancellation" is that the classifying map $X \to BSO(n)$ has a lift through $BFiveBrane(n) \to BString(n) \to BSpin(n) \to BSO(n)$, and $Fivebrane(n)$ at least exists as a group object in sheaves (over the category of manifolds) in homotopy $5$-types. (Is this a sigma-model with bundle-gerbe replaced by a principal bundle for $U(1)$ promoted to a $3$-group?)
Note: We should really be starting with $O(n)$ since a lift of $X \to BO(n)$ through $BSO(n) \to BO(n)$ is the same as equipping $X$ with an orientation.
Anyway, my question is, what exactly is "quantum anomaly cancellation" (perhaps in layman's terms) and what does it have to do with the whitehead tower of $O(n)$?
Also, is there more after $5$-branes?