It is readily apparent from the fossil record that the first limestone was laid down hundreds of millions of years ago, but how long ago was the very first limestone formed? It could be approximately dated by its fossils, but what if it was in the Precambrian and contained no fossils - how could it be accurately dated then?
The mechanism that can lead to inorganic limestone formation is weathering of magmatic carbonatite rocks. I found Siilinjärvi carbonatite complex in Fennoscandia is dated 2.6Ga (Tichomirowa et al, 2013), so Archean. It is the oldest one I found.
Another mechanism that could produce limestones in ancient eons is described by user @BillDOe in this related answer:
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.
The premise comes from this stanford.edu work and is a bit controversial because there are no proofs limestones formed by this mechanism.
M. Tichomirowa, M.J. Whitehouse, A. Gerdes, J. Götze, B. Schulz, B.V. Belyatsky (2013): "Different zircon recrystallization types in carbonatites caused by magma mixing: Evidence from U–Pb dating, trace element and isotope composition (Hf and O) of zircons from two Precambrian carbonatites from Fennoscandia", Chemical Geology, Volume 353, Pages 173-198, ISSN 0009-2541. https://doi.org/10.1016/j.chemgeo.2012.11.004.
Perhaps some of the oldest limestones are the banded Espanola Formation from 2.3 billion years ago. The Oxygenation of the environment happened 2.4 billion years ago. I'll find some references and chemistry info when my web connection works, here are some starter pages.