I've just been using Bott Periodicity to calculate the K-theory of some simple spaces - spheres, torus, and wedge of spheres. The wedge of spheres is interesting.
Given that $\tilde{K}(X \vee Y) = \tilde{K}(X) \oplus \tilde{K}(Y)$ we have that $\tilde{K}(S^n \vee S^m) = \begin{cases} \mathbb{Z} \oplus \mathbb{Z} & m,n \text{ even} \\ \mathbb{Z} & \text{one of } m,n \text{ even}, \\ 0 &m,n \text{ odd.}\end{cases}$
The ring structure is trivial in all cases.
Switching to unreduced K-theory we have $K(S^n \vee S^m) = \begin{cases} \mathbb{Z} \oplus \mathbb{Z}\oplus \mathbb{Z} & m,n \text{ even} \\ \mathbb{Z} \oplus \mathbb{Z} & \text{one of } m,n \text{ even}, \\ \mathbb{Z} &m,n \text{ odd.}\end{cases}$
If both are odd, or one is odd, I can see what the ring structure is. I wonder what it is when they are both even. $K(S^{2n})$ has ring structure $\mathbb{Z}[H]/(H-1)^2$ and $\tilde{K}(S^{2n})$ is generated by $(H-1)$ and has trivial multiplication.
Analogously I guess the ring structure on the wedge is something like \frac{\mathbb{Z}[H,H']}{((H-1)^2,(H'-1)^2)}, where H,H' generate $\tilde{K}(S^n),\tilde{K}(S^m)$, but I guess I am not really sure!
(I am tagging ring theory, since it is probably a standard result in algebra, but as usual feel free to re-tag)