The simplest 8 periodicity related to Bott periodicity and of every day use in physics is the one of the representation theory of the Spin groups.

What is generally called precisely Bott periodicity is about topological K-theory. Topological K-theory is where the charges of D-branes in string theory live. In particular, Bott periodicity has consequences for D-branes. The most obvious one comes from the "complex" Bott periodicity, which is a 2 periodicity and related to the alternance of even-odd dimensions for D-branes in Type IIA-IIB. It is also possible to see the more complicated 8-periodicity i.e. "real" Bott periodicity in the structure of D-branes in Type I. See for example http://arxiv.org/abs/hep-th/9810188