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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 theory is not a triviality - this is more or less equivalent to Bott periodicity.

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 USEFUL cohomology theory.

As pointed in other answers to this question, there is another way to construct K-theory as a cohomology theory - the "infinite loop space machines". You can read in Adams book "Infinite loop spaces" about it - this is one of the most pleasant math books I know, not least because he leaves out all the technicalities. If you go into the details of the infinite loop space machines (May or Segal), it becomes WAY more technical than most the proofs of Bott periodicity, at least if you take the details seriously - it takes May about 80 pages to write everything down. Segals approach looks substantially easier, but that might be due to his very condensed writing.

Using this theory of infinite loop spaces, you get a cohomology theory k^n. This theory is closely related to, but by no means equal to the ordinary K-theory (it is connective, as is any spectrum coming from infinite loop space machines).

If you want to study K-theory seriously, there is absolutely no way around the Bott periodicity theorem: I am not aware of any application of K-theory that can be done without it.The reason is that you cannot compute anything without Bott periodicity.This starts with the first computations (spheres and projective spaces) and goes on to applications like Hopf-invariant one problem, vector fields on spheres, Adams work on Im (J), the Adams conjecture a la Becker-Gottlieb ..., not to mention Index theory.

Last but not least, there are several proofs of B.P. which I found very pleasant to read, for example Atiyah's "Bott periodicity and elliptic operators".

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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.