A time traveler goes back in time (one-way trip, and he does not get to choose the exact destination) to a low-tech society, similar to medieval Europe. His objective is to kickstart an industrial revolution as quickly as possible, and the king of the place he lands in, takes him seriously, wants this to happen and lends active assistance.
He cannot bring any modern equipment back with him, only the knowledge in his head.
He can have any obscure but available knowledge that will help. The exact formula for gunpowder, or the procedure for purifying useful quantities of penicillin? Sure, he can fortunately happen to remember those.
He cannot gain advantage from knowledge of historical events. The reason in this story is that he has gone sideways as well as backward in time, so has landed in a place that resembles medieval Europe, but is not our Europe. But if you prefer, take it as a meta-condition: the question is about the use of knowledge of science and technology, rather than knowledge of historical events, so suppose the time traveler simply happens not to know of any imminent invasions, assassinations or such like.
The king is a practical man. Revelations about the stars being other suns or the nature of atoms are well and good, but what he's actually interested in are ways to improve the security and prosperity of his kingdom. (In other words, the topic is applied knowledge. Pure knowledge for its own sake would be a different discussion.) Sooner is better than later.
I can see how there is enormous advantage to be gained from later developments like rifles, steam engines and mass production. But I'm having difficulty seeing how to gain much practical advantage quickly. It seems likely to take a long time to go from gunpowder to militarily useful firearms, for example.
What is the first innovation that could be developed with future knowledge and local tools and resources, that would provide significant practical advantage?