  Aside from primordial gravitational instability of the cosmological fluid,
various mechanisms have been proposed to generate large-scale structure at
relatively late times, including, e.g., ``late-time'' cosmological phase
transitions. In these scenarios, it is envisioned that the universe is nearly
homogeneous at the time of last scattering and that perturbations grow rapidly
sometime after the primordial plasma recombines. On this basis, it was
suggested that large inhomogeneities could be generated while leaving
relatively little imprint on the cosmic microwave background (MBR) anisotropy.
In this paper, we calculate the minimal anisotropies possible in any
``late-time'' scenario for structure formation, given the level of
inhomogeneity observed at present. Since the growth of the inhomogeneity
involves time-varying gravitational fields, these scenarios inevitably generate
significant MBR anisotropy via the Sachs-Wolfe effect. Moreover, we show that
the large-angle MBR anisotropy produced by the rapid post-recombination growth
of inhomogeneity is generally greater than that produced by the same
inhomogeneity grown via gravitational instability. In ``realistic'' scenarios
one can decrease the anisotropy compared to models with primordial adiabatic
fluctuations, but only on very small angular scales. The value of any
particular measure of the anisotropy can be made small in late-time models, but
only by making the time-dependence of the gravitational field sufficiently
``pathological''.
