The chameleon, or generalizations thereof, is a light scalar that couples to matter with gravitational strength, but whose manifestation depends on the ambient matter density. A key feature is that the screening mechanism suppressing its effects in high-density environments is determined by the local scalar field value. Under very general conditions, we prove two theorems limiting its cosmological impact: (i) the Compton wavelength of such a scalar can be at most ≃1 MPc at the present cosmic density, which restricts its impact to nonlinear scales; and (ii) the conformal factor relating Einstein- and Jordan-frame scale factors is essentially constant over the last Hubble time, which precludes the possibility of self-acceleration. These results imply that chameleonlike scalar fields have a negligible effect on the linear-scale growth history; theories that invoke a chameleonlike scalar to explain cosmic acceleration rely on a form of dark energy rather than a genuine modified gravity effect. Our analysis applies to a broad class of chameleon, symmetron, and dilaton theories.