the complete analytic function
When you start with one function element and continue it in every possible direction, along every possible path, you generate a whole family of elements that are all relatives of the original. The complete analytic function is that ENTIRE family bundled together as one object — all the function elements (equivalently, all the germs) reachable from the starting one by analytic continuation. Instead of choosing a single branch and discarding the rest, you keep every branch at once and treat the multivalued thing as a single, larger entity.
Think of it as the maximal honest answer to 'what is this function?'. For the logarithm, the complete analytic function consists of all the germs Log z + 2 pi i k for every integer k, at every point of the punctured plane — infinitely many branches, all stitched together because you can continue from any one to any other. For the square root, it is the two branches plus or minus sqrt(z), connected because looping around 0 sends one to the other. The complete analytic function does not prefer any branch; the branches are merely the different local views you get after travelling along different paths. Where the function is single-valued (a simply connected piece avoiding the branch points), the complete function restricts to an ordinary holomorphic function; where loops can change the branch, it spreads into several sheets.
This is the conceptual destination of the whole continuation story, and it is exactly what a Riemann surface makes geometric. The complete analytic function is multivalued over the plane, but it becomes a perfectly ordinary single-valued holomorphic function when you regard it as defined on its Riemann surface instead — the surface is built precisely so that each germ sits over its base point as a separate location, and continuation along a path becomes an ordinary walk on the surface. So 'complete analytic function' and 'function on its Riemann surface' are two names for the same idea, one phrased in germs, the other in geometry.
The complete analytic function generated by one branch of sqrt(z) is the pair of germs plus/minus sqrt(z) over each nonzero point. Over a single base point like z = 1 it has two values, +1 and -1; over z = 4 it has +2 and -2. Continuation around 0 swaps the two everywhere at once. Regarded on its two-sheeted Riemann surface, this same object is the single-valued function 'w' where w^2 = z.
Keep every branch at once: the complete analytic function is the full multivalued object, single-valued on its Riemann surface.
The complete analytic function is one object, not a collection of unrelated functions. Calling log 'multivalued' is shorthand for this single connected family of germs — it is single-valued the moment you put it on its proper Riemann surface.