Greyness

The Grey, also known as the grey universe (PROSE: The Infinity Doctors) or greyness (PROSE: Unnatural History) was a horrible emptiness foreseen to fall over the universe in the far future long before the natural end of the universe. It was an alternate timeline to the version of the future with the Needle. (PROSE: The Infinity Doctors)

Nature
According to one account, the phenomenon of the Grey appeared whenever an unstable divergence in history caused a burgeoning alternate universe to try to establish itself by borrowing energy from the original universe. There would not be enough energy in a single universe to support two universes, meaning the process would sap cohesion from time and create a rift in the time vortex which the entire multiverse could dissolve into as if it had never been there. All that would be left would be infinite void of insidious grey mist. (PROSE: Genocide)

Describing the Grey after examining it, the Doctor exclaimed, "There's nothing left. No time, no space, no matter, no energy. The entire universe has gone." (PROSE: The Infinity Doctors)

Predictions
The Doctor was prophesied to lead Gallifrey out of the Time of Darkness. (PROSE: Unnatural History, The Gallifrey Chronicles) According to the Book of Lies, this coming "darkness" would look more like a "greyness". The boy indicated to the Eighth Doctor that Faction Paradox intended to hasten the onset of this greyness. (PROSE: Unnatural History)

The Infinity Doctor theorised that the greyness had been caused by either the Effect or the Time Lords' overreaction to it, unwriting the version of the future with the Needle and replacing it with pure emptiness.

Larna and Omega discussed that the grey universe might have been the result of a time war in their future, possibly even a Time Lord civil war, which damaged parts of time and space beyond repair much as the Time Wars had done. (PROSE: The Infinity Doctors) By some accounts, the War in Heaven occurred in their future (PROSE: The Ancestor Cell, The Infinity Doctors) which left the post-War universe devastated and drained of energy, with whole galaxies uninhabitable.(PROSE: Father Time, COMIC: Miranda) However, parts of the universe had a seemingly natural entropy (PROSE: Hope) and this was the same Needle future which the Infinity Doctor had perceived as alternate to the grey universe. (PROSE: The Infinity Doctors)

Sightings
As part of his attempt to resolve the Rutan-Sontaran War, the Doctor took Sontar and a Rutan ten thousand years into their future with his TARDIS. He intended to show them that their fighting would end in the destruction of both sides; however, when he turned on the scanner, the display was simply grey. Opening the door confirmed that the universe was a sea of grey nothingness. Sontar described it as "grey mist".

The Doctor tested the greyness with his instruments, but there was nothing for him to interpret: it was completely empty, lacking even the miscellaneous virtual particles, photons, and ions present in the most desolate parts of the universe, and it continued forever. This horrified him. (PROSE: The Infinity Doctors)

A loa once showed that post-War universe was actually succumbing to a "sterile greyness" in the era of the Needle. (PROSE: The Story So Far...)

Behind the scenes

 * In Mark Clapham's short story Analysis, published in the 2001 charity anthology Walking in Eternity, followed the amnesiac Eighth Doctor's sessions with a psychoanalyst. When asked about his dreams, the Doctor explained that he often dreams of an empty, grey nothingness.