  We work out the basics of conformal $N=(4,4)$, 2D supergravity in the
$N=(4,4)$, 2D analytic harmonic superspace with two independent sets of
harmonic variables. We define the relevant most general analytic superspace
diffeomorphism group and show that in the flat limit it goes over into the
``large'' $N=(4,4)$, 2D superconformal group. The basic objects of the
supergravity considered are analytic vielbeins covariantizing two
analyticity-preserving harmonic derivatives. For self-consistency they should
be constrained in a certain way. We solve the constraints and show that the
remaining irreducible field content in a WZ gauge amounts to a new short
$N=(4,4)$ Weyl supermultiplet. As in the previously known cases, it involves no
auxiliary fields and the number of remaining components in it coincides with
the number of residual gauge invariances. We discuss various truncations of
this ``master'' conformal supergravity group and its compensations via
couplings to $N=(4,4)$ superconformal matter multiplets. Besides recovering the
standard minimal off-shell $N=(4,4)$ conformal and Poincar\'e supergravity
multiplets, we find, at the linearized level, several new off-shell gauge
representations.
