In this question we learn a great deal about the in-universe warp speed mechanics.
How does the warp scale relate to trans-warp speeds?

Geordi La Forge describes transwarp as twenty times maximum warp and that normal sub-space limitations do not apply to trans-warp variables.

Is this simply 9.99 * 20? Or if not, is there a trans-warp scale?

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    Can't really answer the transwarp portion, but considering by the 24th century, infinite velocity was designated as warp factor ten, it stands to reason, that transwarp is simply a method to achieve speeds above what conventional propulsion was capable of, but yet were still finite. – psubsee2003 Dec 19 '15 at 0:55
  • Ah ha! You perceive my thinly veiled dilemma! – Athena Widget Dec 19 '15 at 0:56
  • You are missing one piece, warp factors are non-linear, so 20*9.99 is not a valid answer for 20x maximum warp. – psubsee2003 Dec 19 '15 at 0:57
  • I always wonder if this is more a problem with the definition of the words 'speed/velocity'. Do they continue to make sense when you are traveling through subspace, or a transwarp conduit, or do they become relative to something else. IE at warp one, they from the point of a planet they are traveling at the speed of light, but what is the speed of the ship, within the warp bubble? – Zoredache Dec 19 '15 at 0:58
  • If you pull at the threads of this particular knot, I've seen some reference to recalibrated warp values corresponding to power input and NOT actual velocity. Then warp 10 becomes the maximum velocity because current warp field mechanics theory states at warp 10 energy input -> infinity. If Trek transwarp uses different theory, then the velocities possible for a given energy input could be higher (e.g. 20x higher) than those for conventional warp theory and the velocity limit of "warp 10" due to infinite energy input goes away. You just have to ignore VOY, which I mostly do anyway, lol. – Jim2B Mar 26 '16 at 17:03

TL;DR: it's complicated.

Memory Alpha has a long section on multiwarp speeds. In TOS and TAS, going beyond warp 10 seems to be perfectly acceptable: the Enterprise reaches warp 14.1 in the TOS episode That Which Survives and warp 22 in the TAS episode The Counter-Clock Incident, while warp 15 is mentioned in the TOS episode The Changeling and warp 36 in The Counter-Clock Incident. According to the book Star Trek: Starship Spotter, the redesignation of warp 10 as infinite speed occurred in the year 2312. The warp factor specifications before 2312 were rated by Starfleet using the Original Cochrane Unit (OCU) warp scale, while warp factors after 2312 use the Modified Cochrane Unit (MCU - not Marvel Cinematic Universe!) warp scale.

But to answer your question about transwarp speed specifically, the best I've found is the following:

In the October 1995 issue of OMNI, science advisor Andre Bormanis stated the idea of warp factors beyond 10 in the alternative future was in a recalibration of the warp scale, as ships had gotten faster. Possibly warp 15 was set to be the transwarp threshold instead, according to Bormanis, and warp 13 in that scale would have been the equivalent of warp 9.95 of the previous scale.

According to Star Trek Encyclopedia (3rd ed., p.555), warp 13 from All Good Things... may also allude to some type of implementation of the Federation transwarp drive technology from VOY: Threshold.

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