In terms of practical algorithms, the Strassen algorithm (O(n^2.8)) is the only one that has runtime advantages for matrix sizes that aren't enormous, and even then, it's not always used because it has two non-trivial costs: reduced numerical stability and more memory space requirements for intermediate results.

It actually doesn't require more memory for intermediate results (see the Strassen reloaded paper). It's more just that it's a ton of work to implement well (even compared to a regular gemm which is already hard), and the benefits only start showing up at pretty large (~4000x4000) matrices.