My money is on Germ Theory. This is a good candidate for a few reasons.

1) Everyone in the modern world has at least some idea about it. Your time-traveler does not need an advanced education to get the idea across and implement it. Keep surgical equipment clean, quarantine sick people, doctors need to wash up between seeing patients, and you're done.

2) It requires no pre-existing technology or equipment. It works at any time period and on any sized society, from a stone-age tribe to Colonial England.

3) It will dramatically increase populations. More people means higher GDP, which tends to mean faster technological growth.

Im going to go in a different direction from most of the current answers, The king would like any methods to improve his kingdom and you want to start an industrial revolution. Don't start with the natural sciences start with the people and policies:

1. A production line drastically improves the rate of production of crafted goods


2. Tax policies - support industries and research


3. Foreign treaties - make it so that you can attract foreign talent


4. Make some level education compulsory for all people


5. Support collaboration over competition for research and have it available to all citizens


6. incentivise innovation

Now you have a baseline to start from, within a few years the revolution should kick off on its own rather than just being based on you and your ideas. Bringing in some useful formulas would be a helpful boost but there is a limit and once you hit that limit you will need to rely on others to keep the innovation going.

Accurate World Map

Being able to draw a highly accurate map of the world would be a huge boon to any seafaring or trading nation. Moreover, knowledge of other civilizations of the time period and their valuable trade goods could kick-start a trading empire.

This doesn't precisely answer the prompt, as it doesn't really improve technology much, but it gives the time traveler the resources to actually make their technological visions a reality.

There isn't really any one invention you could drop in as "the first innovation that could be developed with future knowledge and local tools and resources, that would provide significant practical advantage" that wouldn't really be just a matter of opinion.

from comments.

The printing press with movable type was invented sometime after 1400 so that would be a good one, raising literacy & availability of printed manuals led to a fairly big boost in technical advances.

Crop rotation (four field crop rotation) is another simple idea that produced fairly significant increases in crop yields, that was invented sometime in the 1600's

The concept of microbes & bacteria, boiling drinking water, sterilizing medical instruments etc is another recent one with significant far reaching effects for an early society.

Are all good in there way but (if I'm reading your question aright) you want something that's going to spark further innovation & technological advances in & of itself without much further intervention?

• The steam engine is an easy one with the help of a blacksmith & could have all sorts of implications from agricultural threshing machines to transport, it's the bedrock of our own industrial revolution & once you've any one type of steam engine in relatively common use innovation around this particular bit of tech will rocket.


• The microscope & telescope were invented in the 1500's & 1600's, a lot of science couldn't get started until we had those & it lets you point at those tiny little things in water & say "see this is what we're killing when we boil it so that you don't get cholera"


• The copper zinc battery


• The copper wire electric generator & motor

For those you just need glass-making, blacksmiths, carpenters, pottery & a few cattle (for the batteries, any idea how strong the acid in a cows stomach is?) which are all available in 1200.

Electricity seems like a really important one (alongside microscopes & telescopes) if you want to kick start science really early (which is why I threw in the battery & electric generator / motor), but for them to be anything other than curiosity or novelty items you need something to use them with & I'm at a bit of a loss as to what invention might make use of it that a medieval society would find an immediate & compelling use for so that it would be widely adopted.

Radio might be plausible as a use for electricity with foxhole crystal radios for reception & electricity needed for the transmitting sets, any medieval king would love something like that for his armies, I'm just not sure if the required materials could be sourced in a medieval society.

Watermills & windmills already existed, the Dutch certainly had windmills back then, not so sure about the English.

I think there is a very simple answer to this question that avoids the question of broadness, but in so doing I'm inferring the real question to be 'What knowledge should a time traveller take back to jump start technology?'

The answer is the Maxwell Equations of 1861, and the knowledge that magnetism and electricity are in effect different manifestations of the same thing.

Ultimately, the Maxwell Equations were the theoretical integrations of these two fundamental forces, but from an engineering perspective you don't actually need these equations to figure out how to use the two together. School children do it by hand cranking a wire-wrapped axle in between two magnets, and this is ultimately why the knowledge of electromagnetism is so important; it means you can create energy by turning an axle, and turn an axle by applying electricity. This is a ubiquitous element of our modern technologies by virtue of the fact that we can essentially transmit energy in a portable and useable form to wherever it's needed. That versatility revolutionised all the advancements we gained even through the Industrial Revolution, and in a medieval society, would even give them access to plentiful energy WITHOUT using coal or oil to the extent they ended up doing.

Ironically, this would result in a reverse-steampunk scenario, where fantastical ideas would emerge from electrically driven devices out of mindsets that haven't industrialised yet.

What an industrial revolution WOULD add that isn't currently in place is the ability to scale this technology up. In the first instance, I can see plenty of homes having their own windmills, watermills, etc. to generate enough power for their personal needs. You would end up with literally a cottage industry of electrical generation techniques, all powering personally designed products at different amperages, wattages and the like. So, the reverse-steampunk effect would at least be visually intriguing because of the variety of tools that would all employ electricity in different ways to solve the same problems.

Health information: first aid, personal sanitation, community sanitation, germ theory. That gives you an immediate leg up when it comes to dealing with things like sickness, disease, and public health. A lot of it is simple behaviour and habit and doesn't require a ton of infrastructure.

After that, a lot depends on what level of tech you'd find was already present, but things like the Bessemer process for cheaply producing mass quantities of steel would be good almost any time and weren't technologically too advanced to be created. And once you have cheap steel in mass quantities, all of a sudden you're looking at a whole new world.

Already some really good answers in the comments, I'd just like to suggest an alternative approach:

Communications

While the concept of the Heliograph is simple, it wasn't invented and used in an organised, systemic way until roughly the 1820's. There may have been some isolated cases of using mirrors and shields to pass message, but evidence is spotty at best.

Your time traveller could help the King set up a series of lamp-and-mirror stations through his Kingdom, to rapidly pass messages along. This would increase response time to an invasion by a orders of magnitude, and allow information and intelligence to be more rapidly spread.

Communication would be via Morse Code, and he could also teach them a thing or two about modern cryptography and compression, to further increase their security and efficiency. Come of this could be applied to the King's normal messages as well.

Depending on our time travellers electrical knowledge, he could also attempt to build a bare-bones radio transmitter and receiver. Called a Spark-Grap Transmitter, it basically emits a large pulse of electrical energy that could be detected with a simple crystal radio set up some kilometres away - more if you manage to refine the design. Electrical power could be provided by a simple generator, perhaps turned by a windmill, waterwheel, or handy peasant.

All other answers have the precondition that innovation stays inside the borders of the country.

If this is basically our Earth, then the single biggest question for medieval Europe is who gets to America first - and then who can keep it. Spain became the richest nation in Europe through their gold and silver mines in Mexico. Britain then displaced Spain through state-sponsored piracy and having better ships (and captains).

If your hypothetical country has a coast, then the lesson for medieval and Renaissance Europe is that the people with the best navy win. With better ships, and by essentially controlling the seas, you free up your explorers, traders and settlers/miners to make you money.