After the first few landings of the first stage, Elon Musk said that the first core (landed on land) from the Orbcomm OG2 mission will become a static display outside the SpaceX factory in Hawthorne. (Adding to their collection of the first Dragon to reenter, a Merlin engine hanging in the ceiling. I guess a first stage is too big to hang indoors, maybe in the Tesla factory? Or the BFR factory?).
The GTO mission stage from JCSat 14 mission was the most intense landing to date (being one of the heaviest GTO missions to date) and Elon said they would refire the stage 10 more times after the recovery and use it as a class leader for understanding the ability of the engine to be reused.
As of August 2016, they had test fired it (full duration, 2+ minutes) three times in three days!!!! Which is amazing.
Gwynne Shotwell at the Small Sat 2016 conference at Utah State U in Logan, UT was noted to have said that they will try and slip the JCSAT-14 booster onto the test stand in between the need to test stages for launch until they get to 8-14 test firings.
Thus it seems like they do not actually know the number of firings, but they designed for a large number (planning on reuse, and as detailed in the question, taking lots of firings to launch and land) and plan on stress testing a stage to find out the actual answer.
In terms of practical use, the Block 3 and Block 4 Falcon 9 first stages are being used only 2 times, and then discarded/expended on second launch. The claim is that Block 5 model (due in spring of 2018) will be rapidly and cheaply reusable, up to 10 times before major work. Thus the number of firings per Merlin should be going up quite a bit.
But it will be the penultimate (for now!) Merlin 1D that can perform in this fashion, and it took iterations through 1A->1B->1C->1D and then many tweaks based on 1D experience to get there.
As of mid 2019, the Block 5 has flown a couple of boosters three times, with no fourth flight yet in sight. (Rumoured that the Starlink launches will try to forward the reuse of boosters into the higher reuse numbers. After all, if you are building 4400 satellites, losing a booster with 60 is expensive and annoying, but literally noise in the overall project. This way you let Starlink pay for testing the boosters ability to refly).