In an Earth-like planet with no history of carbon-based life forms, would there be limestone?
I'd have to say, yes. Once life evolved in the world's oceans, the vast majority of limestone formation has been through the organic process. However, before life did evolve, limestone formed through an inorganic process where rain water falling through a CO2-rich atmosphere reacted to form a weak carbonic acid solution that then reacted with calcium-containing minerals to form calcites. This then washed into the world's oceans to form limestone deposits.From gsu.edu:


  Limestone may form inorganically or by biochemical processes.
  Another form of limestone of inorganic origin is oolitic limestone, limestone formed from small spherical grains called ooids.


In an paper from Stanford.edu:


  Carbonic acid dissociated to form hydrogen ions, which found their way into the structures of weathering minerals, and bicarbonate, which was carried down rivers and streams to be deposited as limestone and other minerals in ocean sediments.


So even in a world devoid of carbon-based life, limestone could still form if it had a CO2-rich atmosphere, water (and thus rain), and calcium-containing crustal deposits. Though the paper from Stanford is theoretical, note that this does not mean hypothetical and is (I believe) based on solid science.