Based on <a href="https://mathoverflow.net/questions/80081/what-are-good-examples-of-spin-manifolds">this mathoverlow question</a>, I would like to have a similar list for the case of string manifolds. An $n$-dim. Riemannian manifold $M$ is said to be *string*, if the classifying map of its bundle of orthonormal frames $M \to BO(n)$ lifts to a map $M \to BO(n)<8> = BString(n)$, which is the case if and only if the class $\frac{p_1}{2} \in H^4(M, \mathbb{Z})$ vanishes. There are a lot of models that yield geometric realizations of $String(n)$ either as a topological group (see <a href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Stolz%20AND%20Teichner&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3&mx-pid=2079378">Stolz-Teichner</a>), infinite-dimensional Lie group (see <a href="http://arxiv.org/abs/1104.4288">Nikolaus-Sachse-Wockel</a>) or a  2-group (see <a href="http://arxiv.org/abs/0911.2483">Schommer-Pries</a>). 

> What are enlightening examples of
> string manifolds? What are
> non-examples? When do you have a
> geometric interpretation of the
> obstruction class?

So far, I am aware of the list given at the end of <a href="http://arxiv.org/abs/0810.2131">Douglas-Henriques-Hill</a>. What else is out there?