  We have investigated inflationary model constructed from minimally modified
gravity (MMG) theories. The MMG theory in the form of $f({\bf H}) \propto {\bf
H}^{1+p}$ gravity where, ${\bf H}$ is the Hamiltonian constraint in the
Einstein gravity and $p$ is constant, has been studied. An inflation is
difficult to be achieved in this theory of gravity unless an additional scalar
field playing a role of inflaton is introduced in the model. We have found that
the inflaton with exponential potential can drive inflation with graceful exist
different from the case of Einstein gravity. The slow-roll parameter for both
the exponential and the power-law potentials is inversely proportional to
number of e-folding similar to the case of the Einstein gravity. We also have
found for the scalar perturbation that the curvature perturbation in the
comoving gauge on super Hubble radius scales grows rapidly during inflation
unless $p =0$. For the tensor modes, the amplitude of the perturbations is
constant on large scales, and sound speed of the perturbations can diviate from
unity and can vary with time depending on the form of $f({\bf H})$.
