Observations on power series involving logarithmic singularities

[jaydfox]

I was also thinking that since there is a countably infinite number of singularities, does the natural super-logarithm constitute a meromorphic function? Or does the number of singularities have to be finite?

I'm pretty sure I remember reading that a meromorphic function can have a countably infinite number of singularities, but not an uncountably infinite number.

However, I don't think a meromorphic function can have branches (not least because a branch cut represents an uncountable set), so even the basic logarithm is out.