  I present a new class of nonrelativistic, modified-gravity MOND theories. The
three gravitational degrees of freedom of these ``TRIMOND'' theories are the
MOND potential and two auxiliary potentials, one of which emerges as the
Newtonian potential. Their Lagrangians involve a function of three acceleration
variables -- the gradients of the potentials. So, the transition from the
Newtonian to the MOND regime is rather richer than in the aquadratic-Lagrangian
theory (AQUAL) and the quasilinear MOND theory (QUMOND), which are special
cases of TRIMOND, each defined by a Lagrangian function of a single variable.
In particular, unlike AQUAL and QUMOND whose deep-MOND limit (DML) is fully
dictated by the required scale invariance, here, the scale-invariant DML still
requires specifying a function of two variables. For one-dimensional (e.g.,
spherical) mass distributions, in all TRIMOND theories the MOND acceleration is
a (theory specific, but system independent) function of the Newtonian
acceleration; their variety appears in nonsymmetric situations. Also, they all
make the salient, primary MOND predictions. For example, they predict the same
DML virial relation as AQUAL and QUMOND, and thus the same DML $M-\sigma$
relation, and the same DML two-body force. Yet they can differ materially on
secondary predictions. Such TRIMOND theories may be the nonrelativistic limits
of scalar-bimetric relativistic formulations of MOND, such as BIMOND with an
added scalar.
