- Bott's original proof used Morse theory, which reappeared in Milnor's book Morse Theory in a much less condensed form.
- Pressley and Segal managed to produce the homotopy inverse of the usual Bott map as a corollary in their book Loop Groups.
- Behrens recently produced a novel proof based on Aguilar and Prieto, which shows that various relevant maps are quasifibrations, therefore inducing the right maps on homotopy and resulting in Bott periodicity.
- Snaith showed that $BU$ is homotopy equivalent to $CP^\infty$ once you adjoin an invertible element. (He and Gepner also recently showed that this works in the motivic setting too, though this other proof relies on the reader having already seen Bott periodicity for motivic complex K-theory.)
- Atiyah, Bott, and Shapiro in their seminal paper titled Clifford Modules produced an algebraic proof of the periodicity theorem. EDIT: Whoops x2! They proved the periodicity of the Grothendieck group of Clifford modules, as cdouglas points out, then used topological periodicity to connect back up with $BU$. Relating the periodicity in algebra to the homotopy groups of $BU$ without using topological periodicity was Wood later done by Woodgave a more general discussion of this in Banach algebras and Bott periodicity.
- Atiyah and Bott produced a proof using elementary methods, which boils down to thinking hard about matrix arithmetic and clutching functions. Variations on this have been reproduced in lots of books, e.g., Switzer's Algebraic Topology: Homotopy and Homology.
- A proof of the periodicity theorem also appears in Atiyah's book K-Theory, which makes use of some basic facts about Fredholm operators. A differently flavored proof that also rests on Fredholm operators appears in Atiyah's paper Algebraic topology and operations on Hilbert space.
- Atiyah wrote a paper titled Bott Periodicity and the Index of Elliptic Operators that uses his index theorem; this one is particularly nice, since it additionally specifies a fairly minimal set of conditions for a map to be the inverse of the Bott map.
- Seminaire Cartan in the winter of '59-'60 produced a proof of the periodicity theorem using "only standard techniques from homotopy theory," which I haven't looked into too deeply, but I know it's around.
