Fewest saw cuts to make 27 unit cubes

A $3\times3\times3$ wooden block is to be sawn into $27$ unit cubes. Between cuts you may rearrange and stack the pieces however you like, and each cut is a single straight plane. What is the minimum number of cuts?

$6$. The central unit cube has six faces, all interior to the block, and each cut can create at most one of its faces, so at least $6$ cuts are needed. Six cuts, two in each direction without rearranging, already suffice.