"ever present possibility" is a correct and usable phrase in written English.
You can use it when talking about a recurring possibility that is always present. For example, "The ever present possibility of failure was always at the back of my mind."
It is not an ever present possibility but the given in each case".
We cannot sanction stops justified only by the generalized and ever present possibility that interrogation and inspection may reveal that any given motorist has committed some crime.
They fear an escalation of violence, causing further collateral damage to the economy, foreign investment and tourism, plus the ever present possibility of military intervention.
Justice O'Connor said the court "cannot sanction stops justified only by the generalized and ever present possibility that interrogation and inspection may reveal that any given motorist has committed a crime".
Still, critics are not convinced that enough safeguards are in place, especially with the unpredictable updrafts from the canyons, which the current fires demonstrated, and the ever present possibility of sparks during an intense fire.
They need to be paranoid about the ever present possibility of it happening.
Oh, and there's the ever-present possibility of a gale.
There is also the ever-present possibility of a coal ship running aground on the reef.
Since numbering is considered harassment, detention by the religious police is an ever-present possibility.
The spur I mean is not just the hope of greater rewards but also the ever-present possibility of failure.
I have a sense of the enormous challenge of living a good life, the ever-present possibility of making mistakes.
