  We exhibit subshifts admitting weakly mixing (probability) measures, for arbitrary $\epsilon > 0$, with word complexity $p$ satisfying $\limsup \frac{p(q)}{q} < 1.5 + \epsilon$. For arbitrary $f(q) \to \infty$, said subshifts can be made to satisfy $p(q) < q + f(q)$ infinitely often.   We establish that every subshift associated to a rank-one transformation (on a probability space) which is not an odometer satisfies $\limsup p(q) - 1.5q = \infty$ and that this is optimal for rank-ones. 
Word Complexity of (Measure-Theoretically) Weakly Mixing Rank-One   Subshifts