No. For each countable ordinal $\alpha$, the bounded Baire class $\leq \alpha$ functions on $[0,1]$ constitute a C*-algebra. These C*-algebras are nested, all are contained in $L^\infty[0,1]$, and every semicontinuous function is already in Baire class one.

See http://www.encyclopediaofmath.org/index.php/Baire_classes