In the episode of TNG Where No One Has Gone Before, what speed did the Traveller make the Enterprise go, to pass two galaxies and then afterwards be 1 billion light years away from Federation space?
From Memory Alpha, sourced to the Star Trek: TNG Technical Manual:
In 2364, the Traveler used the energy of his thoughts to move the USS Enterprise-D through space at a speed that registered on instruments as exceeding warp factor 10 and going off the warp scale. (TNG: "Where No One Has Gone Before")
According to Star Trek: The Next Generation Technical Manual (page 55), the ship didn't actually achieve warp 10 or go beyond it, but it did travel at the extreme speed of about Warp 9.9999999996.
In the episode itself, Geordi LaForge says:
LAFORGE: Captain, we're passing warp ten!
But the Manual is more canonical since it was written by the actual technical advisors to the show, whereas LaForge was just going by what the instruments told him, which shouldn't be relied on since the ship was malfunctioning so badly.
In terms of actual speed, the answer is somewhere in the region of
a million million times the speed of light
From the episode transcript:
Captain's log, stardate 41263.2. This will be a rather unusual log entry, assuming Starfleet ever receives it. As I have already informed my crew, a phenomenal surge of power during a warp speed experiment has sent our starship hurtling out of our own galaxy, past another, taking us over two million seven hundred thousand light years in a few minutes.
There are 525,600 minutes in a year, so 2,700,000 light years in a minute would be around 1,400,000,000,000 times the speed of light. Divide by "a few" and it's still, in short, pretty damn fast.