  We describe the topological Hochschild homology of the periodic complex
$K$-theory spectrum, $THH(KU)$, as a commutative $KU$-algebra: it is equivalent
to $KU[K(\mathbb{Z},3)]$ and to $F(\Sigma KU_{\mathbb{Q}})$, where $F$ is the
free commutative $KU$-algebra functor on a $KU$-module. Moreover, $F(\Sigma
KU_{\mathbb{Q}})\simeq KU \vee \Sigma KU_{\mathbb{Q}}$, a square-zero
extension. In order to prove these results, we first establish that topological
Hochschild homology commutes, as an algebra, with localization at an element.
  Then, we prove that $THH^n(KU)$, the $n$-fold iteration of $THH(KU)$, i.e.
$T^n\otimes KU$, is equivalent to $KU[G]$ where $G$ is a certain product of
integral Eilenberg-Mac Lane spaces, and to a free commutative $KU$-algebra on a
rational $KU$-module. We prove that $S^n \otimes KU$ is equivalent to
$KU[K(\mathbb{Z},n+2)]$ and to $F(\Sigma^n KU_{\mathbb{Q}})$. We describe the
topological Andr\'e-Quillen homology of $KU$.
