I must admit I wasn't aware of any that fit the formula!

Could you share any examples?
In my post # 55 at:
https://www.trekbbs.com/threads/tim...lems-in-star-trek.301617/page-3#post-13196805
In some episodes starships seemed to travel no faster than the warp formula indicated.
In "Who Mourns for Adonais?" on the planet Pollux IV Kirk says:
KIRK: On you, Lieutenant! Reject him, and we have a chance to save ourselves. Accept him, and you condemn all of us to slavery, nothing less than slavery. We might never get help this far out. Or perhaps the thought of spending an eternity bending knee and tending sheep appeals to you.
So Kirk think that Pollux or Beta Germinorum would be a long way for a rescue mission to come looking for them. Pollux is only about 33.8 light years from Earth. At warp factor one it would take 33.8 years for a ship from Earth to reach Pollux, at warp factor two it would take about 4.225 years for a ship from Earth to reach Pollux, at warp factor three it would take about 1.25 years, at warp factor five it would take about 0.52 years, at warp factor five it would take about 0.27 years (98.76 days) at warp factor six it would take about 0.156 years (56.9 days), at warp factor seven it would take about 0.098 years (35.99 days) and at warp factor eight it would take about 0.066 years (24.106 days).
So the way Kirk emphasizes how far they are from help, it doesn't seem like Kirk expects any hypothetical rescue ships to come, if at all, much faster than the TOS warp formula indicates.
In "This Side of Paradise"" Kirk & a landing party beam down to Omicron Ceti III:
KIRK: Another dream that failed. There's nothing sadder. It took these people a year to make the trip from Earth. They came all that way and died.
ELIAS: Hardly that, sir. Welcome to Omicron Ceti Three. I'm Elias Sandoval.
If "a year" equals 0.5 to 2.0 Earth years, and if Omicron Ceti is Mira or Omicron Ceti, which was believed to be about 220 light years from Earth, but is now believed to be about 299 light years from Earth, it is simple to calculate the average speed of the voyage. The average speed of the voyage should have been between 110 times the speed of light and 598 times the speed of light.
If the transport ship was limited to warp factor two, the maximum speed of a freighter, it would have traveled 13.75 to 74.75 times as fast as it should have. If the transport ship traveled as fast as warp factor six, the maximum safe speed of the Enterprise, it would have traveled 0.509 to 2.768 times as fast as it should have.
So the transport ship could have traveled many times as fast as the official TOS warp formula if it traveled a low warp, but if it traveled at a high warp factor like warp 6 it didn't have to exceed the official TOS warp formula.
So there are some TOS episodes where spaceships seem to travel at about the official TOS warp formula speeds, and others where spaceships don't have to travel any faster than a few tens of times the official TOS warp formula.
And I can add that "And the Children Shall Lead" was set on Triacus, a planet of Epsilon Indi, a star about 11.87 light years from Earth. If the
Enterprise has reason to visit a star that star should be in the zone where the Federation is exploring space at the time. And it would be consistent with warp drive being as slow as the official TOS warp speeds for a star only 11.87 light years from Earth to still be in the zone of exploration. It would take between 11.87 years at warp one and 0.023 years (8.439 days) at warp eight to reach Epsilon Indi from Earth.
Any place where the
Enterprise patrols the Romulan Neutral Zone is likely to be within the current zone of exploration. In "Whom Gods Destroy" Spock tries to determine which Kirk is the real one and asks:
SPOCK: Fascinating. What maneuver did we use to defeat the Romulan vessel near Tau Ceti?
KIRK 1: Very good, Spock. The Cochrane deceleration.
KIRK 2: Spock, you know the Cochrane deceleration's a classic battle maneuver. Every Starship Captain knows that.
So if the
Enterprise was active near Tau Ceti, Tau Ceti should be within the current zone of exploration. Tau Ceti is 11.905 light years from Earth. According to the official TOS warp scale it would take about 11.905 years to reach Tau Ceti from Earth at Warp one, and about 0.023 years (8.492 days) at warp eight.
In "Spock's Brain" Mr. Spock's brain is stolen and taken to a planet in the nearby star system of Sigma Draconis. If the
Enterprise is near a star that star should be in the current zone of exploration. Sigma Draconis is about 18.77 light years from Earth. According to the official TOS warp scale it would take about 18.77 years at warp one and about 0.036 years (13.39 days) at warp eight to reach Sigma Draconis from Earth.
So those three episodes indicate that the Federation's current zone of exploration is only about 10 to 20 light years from Earth.
But there are also episodes which indicated that stars tens of light years from Earth have already been explored.
In "Conscience of the King" Kirk comments on the exotic costuming of a play:
(On stage, a velvet-clad arm raises a bloody knife, then plunges it again into his sleeping victim. The audience are rapt.)
KIRK: Interesting. An Arcturian Macbeth.
Arcturus is about 36.7 light years from Earth.
"Friday's Child" is set on a planet of Capella or Alpha Aurigae, which thus should be within the Federation's current zone of exploration:
Captain's log, stardate 3497.2. Planet Capella Four. The rare mineral topaline, vital to the life-support systems of planetoid colonies, has been discovered in abundance here. Our mission, obtain a mining agreement. But we've discovered a Klingon agent has preceded us to the planet. A discovery which has cost the life of one of my crewmen.
Capella is about 42.9 light years from Earth.
In "Where No Man Has Gone Before" Dr. Piper introduces Dr. Dehner:
PIPER: Life sciences ready, sir. This is Doctor Dehner, who joined the ship at the Aldebaran colony.
DEHNER: Psychiatry, Captain. My assignment is to study crew reaction in emergency conditions.
Aldebaran or Alpha Taurii is about 65.3 light years from Earth.
In "The Trouble With Tribbles" the star Regulus or Alpha Leonis and its life forms are known to the Federation and to Klingons:
KLINGON: Frankly, I never liked Earthers. They remind me of Regulan blood worms.
CHEKOV: That Cossack.
Regulus is about 79.3 light years from Earth.
Other episodes indicate the zone of exploration should be at least hundreds of light years from Earth.
"The Apple" is at Gamma Trianguli:
Captain's log, stardate 3715.3. While making a routine exploration of the unexplored Gamma Trianguli Six, one of my men has been killed by a poisonous plant.
Gamma Trianguli is about 112.3 light years from Earth, and Gamma Trianguli Australis is about 184 light years from Earth.
"The Deadly Years" begins near Gamma Hydra, which should be Gamma Hydrae:
Captain's log, stardate 3478.2. On a routine mission to re-supply the experimental colony at Gamma Hydra Four, we discovered a most unusual phenomenon. Of the six members of the colony, none of whom were over thirty, we found four had died and two were dying of old age.
Gamma Hydrae is about 133.8 light years from Earth.
In "This Side of Paradise" Omicron Ceti III is visited:
PAINTER: Approaching Omicron Ceti Three, sir.
Omicron Ceti should be about 200 to 300 light years from Earth.
In "the Trouble With Tribbles" planets of the star Spica or Alpha Virginis have been explored:
BARMAN: I don't want any. I told you before, and I'm telling you again I don't want any more Spican flame gems. Thanks to you, I have enough Spican flame gems to last me a lifetime.
Spica is about 250 light years from Earth.
Antares or Alpha Scorpii has been visited:
BARMAN: I don't want any. I told you before, and I'm telling you again I don't want any more Spican flame gems. Thanks to you, I have enough Spican flame gems to last me a lifetime.
JONES: How sad for you, my friend. You won't find a finer stone anywhere. But I have something better. Surely you want some Antarian glow water.
BARMAN: I use that to polish the flame gems.
Antares is about 550 light years from Earth.
Rigel XII is visited in "Mudd's Women":
SPOCK: There's a lithium mining operation on Rigel 12. High-grade ore, I've heard.
KIRK: Location and distance.
SPOCK: Mister Farrell has the course. Less than two day's travel.
KIRK: Make for Rigel 12, Mister Spock.
SPOCK: Rigel 12, Mister Farrell. You have the course.
Rigel or Beta Orionis is about 880 light years from Earth.
And some episodes even indicate that stars thousands of light years from Earth have been reached:
Deneb or Alpha Cygni is mentioned in "The Trouble with Tribbles":
KORAX: That's right, and if I think that Kirk is a Denebian slime devil, well that's my opinion too.
And in "I, Mudd":
MUDD: I sold the Denebians all the rights to a Vulcan fuel synthesiser.
KIRK: And the Denebians contacted the Vulcans.
MUDD: How'd you know?
KIRK: That's what I would have done.
MUDD: It's typical police mentality. They've got no sense of humour. They arrested me.
MCCOY: Oh, I find that shocking.
MUDD: Worse than that. Do know what the penalty for fraud is on Deneb Five?
Deneb is about 2,600 light years from Earth. According to the official TOS warp scale, it would take about 2,600 years to reach Deneb from earth at warp one, and about 5.078 years at warp eight.
At the kind of speed which would enable Harry Mudd to get from Rigel to Deneb in about a year or less many of the closer stars to Earth that have been mentioned above would no longer be within the zone of exploration because it would take only a few days to reach them from Earth.
Thus TOS warp speeds deduced from various episodes range from those that fit the official TOS warp speeds fairly well to those that seem tens and hundreds of times as fast.