As an engineer from the XXI century, how and what would you contribute to the society? What practical benefits would they have for suddenly having you around? What inventions would you be able to accelerate and actually make happen with the resources available?
"Ancient Greece" spans 800 BCE to 500 CE. It spans the wonderful time of Greece, the invasion and assimilation in the Roman Empire, and the creation of Byzantine.
Some light Googling suggests that average male height was around 5'6".
No real steel or iron working existed to speak of, at least by today's standards.
Their number system lacked a zero, and doesn't appear to be positional as ours is.
~
For long effect, I think the most useful things I could contribute would be the basics of integration, derivation, and fixing that damned number system to have a zero and positional numbers.
I'd also try to establish a system of measures that is sensible, because without that engineering is almost impossible.
Lastly, I'd try to encourage a culture of exploring chemistry and mathematics, because without that the advances become secrets become legends become lost.
~
If the Romans are around (Ancient Greece being a long time period), I'm defecting to them immediately.
As a practical matter, showing the workings of a one-time-pad for a really-ghetto-but-computationally-hard-to-crack streaming cypher would be helpful. I'd also demonstrate frequency analysis of captured communications.
I'd then show them movable type so that they can rapidly produce these codebooks--this also makes other things much easier (printing instructions, etc.).
That done, I'd try to get lodestones with compasses working.
If I can find a glassblower, we can try lenses, and if that works, we can make simple telescopes (and then I can try at glasses...I'm there for the long term, after all). Then, I'd introduce the concept of semaphores (the flag type).
So, now we've got the most coordinated army and navy around. Assuming that I haven't been assassinated outright or imprisoned, it's time for the fun part.
Codes build empires, but chemistry and materials science build civilizations.
First, we show the principles of making a still. That'll make everyone happy, and help with sterilizing things.
I'd show them how to make a passable black powder (sulfur, charcoal, and everything you'd need with LeConte's methods being readily available). This will be flashy, but of limited use in either the military or civilian worlds.
More importantly, it creates a supply of nitrates, which we can turn into nitric acid and other useful industrial chemicals. We can use those in turn to make salt bridges and galvanic cells (achievement unlocked: electrochemistry), because we have copper and zinc. That's for later.
So, big problem is, we need iron. In the north, the Germans are sitting on a bunch of coal and iron deposits. We show them how to mine it, we open trade, and then we start iron production.
We also need to invent lathes, and from there, basic machining tools. That will eventually let us do things like draw wire, turn screws, and basically do semi-modern manufacturing.
With iron and copper, we can make pressure vessels and exchange tubes. That in turn lets us make boilers, and in turn, steam power. Steam drives can run factories. The rest is left as an exercise for the reader.
~
The two biggest problems (assuming we can even communicate enough to start) are:
1. Not getting killed off or burned as a witch.
2. Not dying from the myriad of terrible untreatable (for then) diseases.