Formation of limestone without living organisms?
Oolites are limestones that are usually considered as in-organic, although they may have bits of shell/etc in them.  These form by the precipitation of calcium carbonate around particles (sand, broken shell, etc) with a process comparable to that of an oyster.

Some oolite references:

http://www.sandatlas.org/2012/09/oolite/

http://en.wikipedia.org/wiki/Oolite

Recently I saw a reference that limestones do not deposit when Fe2+ is present. Unfortunately I can't find the reference now. If this is the case, then for the Earth, limestones require the indirect presence of life. The evolution of cyanobacteria created free oxygen in the atmosphere and the dissolved in the oceans. Hence all dissolved iron was oxidised from Fe2+ (a green colour) to Fe3+ (orange colour).