I opine that everyones favorite should be the proof using Toeplitz operators (as described in Higson-Roe); essentially due to Atiyah (in ''Bott periodicity and the index of elliptic operators''). In the commutative case, it is the cleanest and most memorable proof. It gives an explicit homotopy inverse to the Bott map as a map $\Omega U \to Z \times BU$ spaces). It is fairly elementary. It is directly linked to the Toeplitz index theorem (that relates the most elementary topological invariant, the winding number, to an index). The Toeplitz index theorem is, moreover, one of the building blocks of the Atiyah-Singer index theorem. The proof can be generalized to the Real case. It sets the stage for Cuntz' proof of periodicity in $KK$-theory. There is one drawback that I am willing to take serious: there is no straightforward generalization to the Clifford-linear case.

I opine that everyones favorite should be the proof using Toeplitz operators (as described in Higson-Roe); essentially due to Atiyah (in ''Bott periodicity and the index of elliptic operators''). In the commutative case, it is the cleanest and most memorable proof. It gives an explicit homotopy inverse to the Bott map as a map $\Omega U \to Z \times BU$ spaces). It is fairly elementary. It is directly linked to the Toeplitz index theorem (that relates the most elementary topological invariant, the winding number, to an index). The Toeplitz index theorem is, moreover, one of the building blocks of the Atiyah-Singer index theorem. The proof can be generalized to the Real case. It sets the stage for Cuntz' proof of periodicity in $KK$-theory. There is one drawback that I am willing to take serious: there is, as far as I can see, no straightforward generalization to the Clifford-linear case.