  We investigate nonlinear elliptic Dirichlet problems whose growth is driven
by a general anisotropic $N$-function, which is not necessarily of power type
and need not satisfy the $\Delta_2$ nor the $\nabla _2$-condition. Fully
anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional
framework associated with these problems. Minimal integrability assumptions are
detected on the datum on the right-hand side of the equation ensuring existence
and uniqueness of weak solutions. When merely integrable, or even measure, data
are allowed, existence of suitably further generalized solutions - in the
approximable sense - is established. Their maximal regularity in
Marcinkiewicz-type spaces is exhibited as well. Uniqueness of approximable
solutions is also proved in case of $L^1$-data.
