  Recently there has been interest in extending ghost-free massive gravity,
bi-gravity, and multi-gravity by including non-standard kinetic terms and
matter couplings. We first review recent proposals for this class of
extensions, emphasizing how modifications of the kinetic and potential
structure of the graviton and modifications of the coupling to matter are
related. We then generalize existing no-go arguments in the metric language to
the vielbein language in second-order form. We give an ADM argument to show
that the most promising extensions to the kinetic term and matter coupling
contain a Boulware-Deser ghost. However, as recently emphasized, we may still
be able to view these extensions as effective field theories below some cutoff
scale. To address this possibility, we show that there is a decoupling limit
where a ghost appears for a wide class of matter couplings and kinetic terms.
In particular, we show that there is a decoupling limit where the linear
effective vielbein matter coupling contains a ghost. Using the insight we gain
from this decoupling limit analysis, we place an upper bound on the cutoff for
the linear effective vielbein coupling. This result can be generalized to new
kinetic interactions in the vielbein language in second-order form. Combined
with recent results, this provides a strong uniqueness argument on the form of
ghost-free massive gravity, bi-gravity, and multi-gravity.
