Superscience is a special TL that is outside the normal GURPS TL scale.
What is Superscience in GURPS 4e?Edit
'“Superscience” technologies violate physical laws – relativity, conservation of energy, etc. – as we currently understand them. (...) By definition, it is impossible to set a firm TL for superscience – we might discover faster-than-light travel tomorrow, a thousand years from now, or never. Equipment TLs are always debatable, but superscience TLs are arbitrary. To reflect this, the rules give the TL of superscience developments as “^” TLs are arbitrary. To reflect this, the rules give the TL of superscience developments as “^” instead of a number.' The GM is free to assign such innovations to any TL. To note a superscience invention that appears at a specific TL in a particular game world, put a “^” after its TL; e.g., “TL3^” for a TL3 superscience item. 
This is why broadcast power has two totally different TLs: TL6^ (Infinite Worlds, Gernsback) and TL10^ (Ultra-Tech).
The problem is the superscience category didn't exist in 3e and so books like GURPS Steampunk and Steamtech used TL(x+y) to denote both divergent TL (vanilla TLx+y) and superscience TL (TLx^ or TL(x+y)^). So you would have a totally realistic drug like Aspirin and a superscience drug like Atavismine (a devolution drug; effectively what happens in Monster on the Campus (1956). Likely inspired by the Strange Case of Dr Jekyll and Mr Hyde novel.) both listed as TL(5+1) in GURPS 3rd edition.
Unfortunately superscience is associated with the specific TL it appears even if the overall TL has moved on. Things are complicated by the fact the dates given for when TLs begin should be viewed as rough approximations not hard and fast dates. This makes it difficult to figure out just where some of the borderline cases reside.
For example, the Nautilus of 20,000 Leagues Under the Sea (advanced submarine, GURPS Steampunk pg 80) uses cinematic sodium/mercury batteries which it must return to base to recharge. In GURPS 3e advanced submarine was listed TL(5+1) and but it is unclear if under GURPS 4e it should still be that TL. Moreover there as been at least on retcon of the Nautilus to having nuclear power (TL7) making advanced submarine TL(5+2) or a TL7 oddball. This means that depending on how one views advanced submarine it could be TL5^, TL(5+1), or TL7 (or should that be TL(5+2)^?).
Compounding matters is that superscience is also applied to 'realistic' tech that is developed far ahead of when it "should" as demonstrated by the Britannica-5 setting (see below).
Alfheim: TL3+magic because no "technology" (presumably including alternate TLs) above TL3 works or quickly falls apart.
Azoth-4: China is at TL2^ with everybody else at the TLs they were in 1010.
Azoth-7: TL(4+2) (TL4^ space and weapons). This demonstrates how taking the TL associated with superscience literally produces really silly results. If taken as actually presented, Astronomy on the ground is TL(4+2) in Azoth-7 but TL4^ on a space ship which incurs penalties when going from one to the other.
Britannica-5 (GURPS Infinite Worlds-Lost Worlds pg 6) has antimatter bombs but is put at TL5^ rather then the TL10 that the Basic set says such technology is "normally" developed at.
In fact, Britannica-5 shows another aspect of superscience: much earlier development of technology that would "normally" happen far latter. So a TL7 computer that is somehow invented at TL4 would a TL4^ device not TL7. More over its Infinite Worlds description is totally useless at figuring what the TL is: "Britannica-5 is a steampunk reality absorbed into Zone Green a decade ago." (that's it) Per GURPS Steampunk Britannica-5 would likely be called TL(5+1) rather then the TL5^ it properly got.
Cyrano (GURPS Infinite Worlds-Lost Worlds): TL(5+4) (space travel, TL4^); effectively same situation as Azoth-7.
Etheria (GURPS Infinite Worlds-Lost Worlds pg 5): TL(5+1) (etheric spacecraft, TL5^) - updated version of GURPS Steampunk world which was originally simply TL(5+1). It is stated that it has "subtly askew physical laws" (Venus and Mars are habitual without spacesuits).
"Falkenstein (Q4, current year 1873), a strange blend of sorcery and TL(5+1) technologies has raised Bavaria to near-great power status; elves, dragons, and other nonhuman races abound. (See GURPS Castle Falkenstein for further details.)" (GURPS Infinite Worlds-Lost Worlds pg 6). Magic has existed here and has effected technology for a long time.
Jotunheim (GURPS Infinite Worlds-Lost Worlds pg 10): cosmic energy effectively replaces atomic power and it is called TL7^.
Muspellheim (GURPS Infinite Worlds-Lost Worlds pg 11) is a mixture of TL(6+3) and TL7^ thanks to reverse-engineering of UFO technology.
Merlin-1: TL(7+1) magitech. Likely has a mixture of TL TL7^ and TL(7+1)^ due to the magic.
Roma Arcana (Rome-6): TL2 with mythic and magical elements.
Yrth (GURPS Banestorm): 3-5 (gunpowder suppressed; some TL3/6+ elements from newcomers) However even with gunpowder suppressed magic allows the production of TL5-6 equivalents.
For example, Shape Metal combined with coal and water could easily produce steam power which would totally change the balance of power. Many other spells could be used in this manner to produce TL5-6 items, and you really can't hand wave that with the idea that the mages wouldn't allow such tech because it would threaten their power.
Anyone who has looked at the Gilded Age can see how rubbish that idea is as the Robber Barons had great amounts of political power with no magic. Yrth mages would gain even more power so unless they are idiots something else (or someone else) is slowing technological development on Yrth.
Dead End divergent TL devicesEdit
Then there is the technology that didn't go anywhere but worked. For example the heavier than air steam powered flying machine of 1848...though flying would be generous. Hopping might be a better term. A better version appeared in 1874 but that was more a powered glider then what we would consider an aircraft. In 1890 there was another short flight by a steam powered aircraft, the 1894 flight of another steam powered aircraft had a trip of several hundred feet. It would be 1933 before Besler made a steam powered plane that actually flew. So depending on how it was achieve a heavier than air steam powered flying machine could be TL5^ or a TL(5+1) invention.
Examples on using superscienceEdit
It existed...but didn't actually workEdit
These are all real world discoveries, inventions, or methods that didn't work because (unknown to the people at the time) they violated physical laws or were based on flawed understanding. In a superscience world these could have actually existed or worked as presented.
Electropathic Belt (TL6): In the real world this didn't do anything. In a superscience setting "it increases the wearer's HT to 10, if it was below that level". (Steamtech 31)
N-Rays ("discovered" in 1895): they were actually the result of experimenter bias. In a superscience world they could actually exist.
Phrenology (lampooned in Bowery Bugs; TL5): the shape of, size of, and bumps on the skull denotes intelligence and certain personality traits.
The Tempest Prognosticator (TL5): appeared in the Crystal Palace exhibition of 1851. It was supposed to predict the weather. In the real world it, at best, amounted to little more then a very crude barometer but in a superscience setting it can give a +2 to Meteorology. If the GM wants to whole hog with the superscience it can predict earthquakes as well. (Steamtech pg 51)
Lord Kelvin's Water-Drop Electrostatic Generator (TL5): In the real world it has been relegated to the demonstrate the principles of electrostatics in physics education. In a superscience world it could be more useful then Volta's Pile. (Steamtech pg 52)
Cheirometer (TL(5+1)): A limited version of the Bertillon System of Criminal Identification system, which served the same function as fingerprints by "the meticulous measurement and recording of different parts and components of the human body" and still survives in the form of mug shots. This focuses on the measurements of only the hands. (Steamtech pg 58)
Nearly all pre 1940s science fictionEdit
Around TL6^ with the occasional higher "realistic" TL appearing.
Star Trek Original Series (TOS)Edit
See GURPS Prime Directive for details.
TL(6+3)^ (transtator: TL6^)...maybe.
EU Star WarsEdit
As with Star Trek much of Star Wars tech is superscience.
According the EU tie ins the power source of the Death Star is an atomic reactor with cinematic power output. This all puts EU Star Wars at TL8^
A Victorian-era space-faring setting similar to Etheria. It is a heady mixture of TL6^, TL(5+1), TL(5+1)^, and the occasional TL(5+2)/(6+1) and TL(5+2)^/(6+1)^ gizmo.
The AD&D supplement Ravenloft: Domains of Dread introduced the concept of Cultural Levels which roughly equate to GURPS TLs:
Iron Age; Classical: TL2
Dark Age; Early Medieval; Medieval; Chivalric: TL3-4
"Masque of the Red Death" (Ravenloft): TL5-6
The presence of magic does mess up the TL scale a bit as if it is common enough then by Niven's Law ("Any sufficiently advanced magic is indistinguishable from technology.") the effective TL is changed.
GURPS 4e Fantasy pg 66 give examples using the GURPS magic system but the principle is the same.
As a result the Spelljammer setting would sit at TL4 (space TL4^ thanks to its interplanatary travel) and perhaps be going toward TL(4+4)^ at a good clip, if it is isn't effectively there already.
Azeroth (World of Warcraft)Edit
Mixture of TL3^ - TL6-7^.
Castles are TL3 in design, TL4-5 gunpowder weapons are all over the place, Dwarves can build TL5-6 tanks, and there the Gnomes and Goblins whose TL6-7 inventions are just as likely to blow up in your face as work as intended. In some cases magic powers these machines devices or they violate physical laws hence the "^".