There is a definition of $K^n$ for positive $n$, without Bott periodicity. This approach goes back to Karoubi, and you can find it in his book "K-theory". The definition (for both, positive and negative $n$) uses Clifford algebras, but no Bott periodicity (Karoubi uses his new, more algebraic and direct definition to prove Bott periodicity). On the other hand, without Bott periodicity, topological $K$-theory is not very interesting. Also, the identification of Karoubi's negative $K$-theory with the ordinary is not a triviality.

Another way to phrase the problem: To define $K^{-n}(X)$, you can take the n-fold suspension of $X$ or the n-fold loop space of $Z \times BU$. If you wish to define $K^n (X)$, you need a space $Y_n$ whose $n$-fold loop space is $Z \times BU$. So what you need to know is that $Z \times BU$ is what topologists call an "infinite loop space", see Adams nice book with the same title. This is what Bott periodicity does, and it is absolutely crucial to define $K$-theory as a cohomology theory.