  For a fixed analytic function $g$ on the unit disc $\mathbb{D}$, we consider
the analytic paraproducts induced by $g$, which are defined by $T_gf(z)=
\int_0^z f(\zeta)g'(\zeta)\,d\zeta$, $S_gf(z)= \int_0^z
f'(\zeta)g(\zeta)\,d\zeta$, and $M_gf(z)= f(z)g(z)$.
  The boundedness of these operators on various spaces of analytic functions on
$\mathbb{D}$ is well understood. The original motivation for this work is to
understand the boundedness of compositions of two of these operators, for
example $T_g^2, \,T_gS_g,\, M_gT_g$, etc. Our methods yield a characterization
of the boundedness of a large class of operators contained in the algebra
generated by these analytic paraproducts acting on the classical weighted
Bergman and Hardy spaces in terms of the symbol $g$. In some cases it turns out
that this property is not affected by cancellation, while in others it requires
stronger and more subtle restrictions on the oscillation of the symbol $g$ than
the case of a single paraproduct.
