the universal cover
Among all the covering spaces sitting over a base space, one is the most spread-out of all: the universal cover. It is the covering that is itself simply connected — it has no holes left, no loops that cannot be shrunk to a point. Intuitively, you keep unrolling the base, undoing every loop, until there is nothing left to unroll. Every other connected covering of the base can be obtained from the universal cover by folding it back up, which is why it is called universal: it sits on top of all the others and projects down onto each of them.
The simplest example is exactly the logarithm's spiral ramp. The punctured plane has one essential loop (around the puncture), and unrolling that loop forever gives the infinite spiral with no top, no bottom, and no holes — and that spiral is the universal cover of the punctured plane, realised concretely by the map exp from the whole plane down to the punctured plane. On the universal cover, the multivalued logarithm becomes a single-valued function (it is just the inverse of exp), because once every loop is unrolled there is no loop left to change the branch. More generally, any multivalued analytic function defined by continuation on a region becomes single-valued when pulled up to the universal cover of that region — this is the strongest form of the monodromy idea.
The universal cover is the deep engine behind the uniformization theorem, the grand classification of Riemann surfaces. That theorem says every simply connected Riemann surface is conformally one of just three models: the Riemann sphere, the complex plane, or the unit disk (equivalently the upper half-plane). Since the universal cover of any Riemann surface is simply connected, it must be one of these three — so every Riemann surface is one of these three model spaces folded up by a group of symmetries. This is how the torus turns out to be the plane folded by a lattice, and how every surface of genus at least 2 turns out to be the disk folded by a group of hyperbolic motions: the universal cover supplies the geometry, and the folding recovers the surface.
The torus is the quotient of the plane by a lattice: its universal cover is the whole complex plane, and the covering map wraps the plane around the doughnut, with the lattice of translations doing the folding. The punctured plane's universal cover is the logarithm's spiral ramp (the plane via exp). In both cases the cover is simply connected and the messy multivaluedness downstairs becomes single-valued upstairs.
Unroll every loop until nothing is left: the universal cover is simply connected, and every other covering folds out of it.
The universal cover is simply connected by definition, and it exists for any reasonably nice (locally simply connected) space. Via uniformization it is always one of just three conformal models — sphere, plane, or disk — which is the whole reason that theorem classifies all Riemann surfaces.