"endless pursuit" is a correct and usable phrase in written English.
You can use it to refer to an activity or objective that someone is attempting to continuously and relentlessly complete, with little or no success. For example, you could say "The explorer continued his endless pursuit of the lost treasure".
What is this endless pursuit of millenials about?
The chairman, Billy Payne, says the club is driven by "the endless pursuit of perfection".
Our meaning and purpose does not lie in the endless pursuit of money nor in an allegiance to abstract principles.
But the endless pursuit begins to feel a little pointless and, even with a couple of false-bottom shocks, repetitive.
He dismisses them as either rapacious and untutored exploiters or as moneyed dilettantes engaged in the endless pursuit of pleasure.
The endless pursuit of economic growth is making us unhappy and risks destroying the Earth's capacity to sustain us.
It's a world laden with echoes from the ancient British myth-kitty, and the various raids on it by previous writers: Gawain is here, aged and in endless, fruitless pursuit of Querig, the mist-breathing dragon, bringing to mind TH White's hapless King Pellinore and the Questing Beast.
You probably put on a fake smile in public and forge ahead with busy, endless activity in pursuit of success and the American dream of more, more, more, when all you really want is just to go home, crawl into bed and watch Netflix all day.
As he saw it, his wasn't a glamorous life of artistic pursuits, endless parties and trendcasting.
Long confined to the serious appreciation of high art and classical music, it is now applied to an endless cascade of pursuits.
The destination's many facets--from the broad beaches of Venice and the sea-kissed cliffs of Malibu in the West to the granitic peak of Mt. Baldy and the arid hills in the East--as well as its different cultural textures guarantee an endless array of pursuits, all clamoring for everyone's attention.
