I will present recent joint work with Austin Anderson and Mirjana Jovovic on $[\Phi_t,X]$, the maximal space of strong continuity for a semigroup of composition operators induced by a semigroup $\{\Phi_t\}_{t\geq 0}$ of analytic self-maps of the unit disk, when $X$ is $BMOA$, $H^\infty$ or the disk algebra. In particular, we show that $[\Phi_t , BMOA]$ is not $BMOA$ for all nontrivial semigroups. We also prove, for every semigroup $\{φ_t\}_{t≥0}$, that $\lim_{t\to 0^+} \Phi_t(z) = z$ not just pointwise, but in $H^\infty$ norm. This provides a unified proof of known results about $[\Phi_t,X]$ when $X \in \{H^p, A^p, B_0, VMOA\}$.