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In "That Which Survives" the crew of Kirk's Enterprise unceremoniously hit Warp 14.1 I don't recall a higher warp ever being achieved.

I am aware that there are other questions and answers regarding a re-imagining of Warp wherein Warp 10 is basically infinitely fast, but was there any episode or Star Trek series wherein they stated a higher Warp than 14.1 was achieved?

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2  
I'm reasonably sure this contains the answers you're looking for, but the way you've phrased it (specifically whether a speed of more than warp 14.1 was ever called) makes it not a dupe. – Valorum Mar 25 at 23:20
    
@Richard Thanks, Richard. Yea, I read that post about a week ago and then I just watched this episode, so I thought I would ask about it very carefully. – Hack-R Mar 25 at 23:22
    
Does this answer your question? – Rand al'Thor Mar 25 at 23:31
    
@randal'thor I think so. Your answer below is more direct and also more readable, so I think it's a valuable contribution though. – Hack-R Mar 26 at 0:21
    
The out of Universe explanation is that the writers were lazy and used what ever number sounded impressive. Any discussion of "warp scale recalibration" is retcon by Trek creators after they discovered what a big mess the writers from different shows made. I assume most individual writers were self-consistent but remember multiple writers, multiple shows, over multiple decades - and most of these paid no attention to what the others did. – Jim2B Mar 26 at 16:52
up vote 9 down vote accepted

Yes, higher warps than 14.1 have been recorded.

In the TAS episode The Counter-Clock Incident, the Enterprise encounters a ship travelling at warp 36. From the transcript (emphasis mine):

SPOCK: Captain, sensors are picking up an unidentified object travelling at an incredible speed, presently on collision course with the Enterprise.
KIRK: Put the ship on Red Alert. How fast is it travelling, Mister Spock?
SPOCK: At a speed that should be impossible to achieve. Something on the order of warp thirty six.
APRIL: No natural object has ever been recorded as travelling at that speed.
SPOCK: I believe it is not a natural object, but a ship.

Later on, the Enterprise itself accelerates to a speed of at least warp 22:

KIRK: Mister Arex, what is our speed?
AREX: Now at warp eleven, Captain.
[...]
AREX: Still increasing speed, Captain. Warp fourteen. Warp fifteen.
[...]
SPOCK: Our speed is now warp twenty, and the other ship will contact the nova in one minute fifty eight point three seconds.
[...]
SPOCK: We are up to warp twenty two and increasing, Captain. We shall have forty two point eight five seconds to correct our course after the alien ship enters the nova.

Even if you don't want to count TAS as canon, the answer is still the same just within TOS. Although the Enterprise itself isn't recorded as travelling at such speeds, they encounter a bolt of energy travelling at warp 15 in the episode The Changeling:

SULU: Captain, shields just snapped on. Something heading in at multiwarp speeds.
KIRK: Evasive manoeuvres, Mister Sulu.
SPOCK: An extremely powerful bolt of energy, Captain.
[...]
KIRK: Mister Spock, speed of those bolts.
SPOCK: Approximately warp fifteen, Captain.
KIRK: Then we can't out run them.

For more details and discussion, you can find a long section on multi-warp speeds at Memory Alpha.

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How do these speeds compare to TNG-era speeds? They recalibrated the scale, so we can't just compare them numerically. – Kevin Mar 26 at 1:22
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@Kevin That Memory Alpha link goes into a lot of detail on these issues. The redesignation of warp 10 as infinite speed occurred in the year 2312, when Starfleet started using the MCU warp scale rather than the OCU one. – Rand al'Thor Mar 26 at 1:26

(Following answer does not treat TAS as canon)

All Good Things had an alternate future Enterprise D travel at warp 13, which should be faster than warp 15 from TOS (5165.88 *c and 3,375 *c, respectively), given the scale recalibration, if we assumed a linear function for warp (which it isn't, so warp 13 is actually much much much faster than that). However (as reported in the Memory Alpha article), warp 13 in that future would have been regarded as warp 9.95 in regular-time TNG scale...which is, unfortunately, slower than the TOS 15, as warp 9.975 is only 3053 *c (according to VOY). VOY's tech manual states that 9.99 is 7912 times the speed of light, which would be 19.92~ in TOS speeds.

So! What this does mean is, according to the speeds stated in the VOY tech manuals, the Enterprise D has reached the highest warp speed yet when it was thrown through space by the traveler, traveling at Warp 9.9999999996 (according to the TNG tech manual). And that's it? The fastest warp is less than 10, using a different scale and exponential function that I can't find.

However! Nothing (afaik) on the show in TNG or VOY was stated at traveling above 9.98~ish (which should be faster than TOS 15), so @rand al'thor has the best answer as the lightning bolt from TOS.

For reference:

TOS: v = (warpFactor to the power of 3) * c

TNG: v = if warpFactor <= 9 then (warpFactor to the power of 10/3) * c

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"Nothing (afaik) on the show in TNG or VOY was stated at traveling above 9.98~ish" - What about the VOY episode "Threshold" where Tom Paris breaks the warp 10 barrier? – Remy Lebeau Mar 27 at 2:06
1  
I have a blindspot around that episode. Here is why: scifi.stackexchange.com/questions/63134/… – n_b Apr 1 at 15:02

I don't recall them mentioning a warp speed, but TNG 1x06 "Where No One Has Gone Before" is worth a look: "On the bridge, La Forge tells the captain they are passing warp 10" and "Data later says that their velocity is off the scale." (both from Memory Alpha, "Where No One Has Gone Before (episode)" episode guide)

During the same episode, they travel to "1 billion light years from the Milky Way Galaxy in the other direction" (same page), though without a duration, no speed can be calculated. I think from context, a reasonable upper limit of the time may be estimated, and from that a lower limit of the speed may be calculated. For example, if you say the last mentioned trip took no more than 8.7 hours (1/1,000 of a year, almost certainly from viewing the episode), then they travelled at 1 trillion times the speed of light; that would be warp 10,000 at TOS speed.

Another place to look would be all the episodes with John de Lancie as "Q" - not being limited by those laws of physics, though I think Q didn't so much "travel", he was just "there". Also the pilot to Voyager - I don't know is a specific speed is mentioned, but I doubt it would be as fast as in TNG 1x06.

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