The most natural examples are irrational rotations. Are there other examples that are fundamentally different from irrational rotations? By the way, if $T$ is totally ergodic, but not weakly mixing, then there must be an eigenvalue of $U_T$ (the natural unitary operator associated to $T$) that is an irrational multiple of $2\pi$; this gives at least some connection between transformations that are totally ergodic but not weakly mixing, and irrational rotations.
To be extremely specific, I would like to construct (if possible) a rank-1 transformation with bounded cutting parameter that is totally ergodic but not weak mixing. I'm looking for examples which will help me understand the ways in which a transformation can have totally ergodicity but not weak mixing, and then use that understanding in developing the construction.