I guess it will scale up to such a degree that we will not be able to count all the transactions. There will be more meshes of blockchain than the stars in the sky 
Quote
By measuring the number and luminosity of observable galaxies, astronomers put current estimates of the total stellar population at roughly 70 billion trillion
To reach that every human in this world would have to make 10 trillion transactions.
If we give an estimate of 1 million tx in a lifespan we're going to have to wait for 10 million generations in order to reach that.
Hope I got the math right...I have the feeling something is off.
But is it possible to stop one day ?
Yes, the moment there is not a single miner left mining blocks the chain stops.