Observe that we only look at consecutive circles. Suppose we have a disk chain which ends up at the same point. Although it ends at the same point, the coherence condition of the chain is that consecutive circles agree. The first and last circles might be "far away", so there is no reason that they should agree. Therefore we cannot glue all circles together to get a holomorphic function. It also means that two disk chains starting with $C_0$ and ending at a point $z$ may have different values on the last circle.

If this wasn't the case, then we could define the natural log in a circular region, which is absurd.

This problem is called Monodromy, and will be addressed later.