  The dark ages 21-cm signal is a powerful tool for precision cosmology and
probing new physics. We study two non-standard models: an excess radio
background (ERB) model (possibly generated by dark matter decay) and the
millicharged dark matter (mDM) model. These models were inspired by the
possible EDGES detection of a strong global 21-cm absorption during cosmic
dawn, but more generally they provide a way to anticipate the potential
discovery space. During the dark ages the 21-cm global signal in the ERB model
reaches a saturated form for an amplitude $A_{\rm r}=0.4$, where $A_{\rm r}$ is
the radio background intensity at cosmic dawn relative to the cosmic microwave
background. This amplitude is one-fifth of the minimum required to explain the
EDGES signal, and corresponds to just 0.1% of the observed extragalactic
background; it would give a signal that can be detected at 5.9$\sigma$
significance (compared to $4.1\,\sigma$ for the standard signal) and can be
distinguished from the standard (no ERB) signal at $8.5\,\sigma$, all with a
1,000 hr global signal measurement. The 21-cm power spectrum has potentially
more information, but far greater resources would be required for comparable
constraints. For the mDM model, over a range of viable parameters, the global
signal detection significance would be $4.7-7.2\,\sigma$, and it could be
distinguished from the standard at $2.2-9.3\,\sigma$. With an array of global
signal antennas achieving an effective 100,000 hr integration, the significance
would be $10\,\times$ better. Our analysis helps motivate the development of
lunar and space-based dark ages experiments.
