  We introduce the notion of log-Riemann surfaces. These are Riemann surfaces
given by cutting and pasting planes together isometrically, and come equipped
with a holomorphic local diffeomorphism to C called the projection map, and a
corresponding flat metric obtained by pulling back the Euclidean metric. We
define ramification points to be the points added in the metric completion of
the surface with respect to the induced path metric; any such point has a
well-defined order $1 \leq n \leq +\infty$ such that the projection map
restricted to a small punctured neighbourhood of the point is an $n-to-1$
covering of a punctured disk in C. We prove that simply connected log-Riemann
surfaces with finitely many ramification points are biholomorphic to C and the
uniformization, with respect to the distinguished charts on the surface given
by the projection map, is given by an entire function of the form $F(z) = \int
Q(z)e^{P(z)} dz$ where $P, Q$ are polynomials of degrees equal to the number of
infinite and finite order ramification points respectively. We also develop an
algebraic theory for such log-Riemann surfaces, defining a ring of functions on
the surface with finite values at all ramification points, such that the ring
separates all points including the infinite order ramification points.
