Transport: Pockets or woven bags must rate pretty high as Enablers; removing basic limits on foraging, allowing eating to be deferred to safe locations (including weaning children), creating trading possibilites, creating the need for other inventions like measurements, counting, drawing/mapping (more berries like these over there). Apes don't need advanced technologies because they can only carry a little at a time and so food is never an abstract concept, separated from its source.
Probably language was invented around the meal table - it was the only place where it was safe to make any noise.
No doubt at all in my mind. Without it, we'd still be just another great ape eating berries and the occasional animal. Nothing much more advanced than basics like stone tools, levers, rollers or wheels and simple shelters seems likely.
The moment I saw the question, I scanned the answers looking for it.
It's encouraging to me to see so many of us in engineering see it, because it seems to me that most other groups overlook it. I think it demonstrates that engineers see a bigger picture than most people.
Of course there are dozens of close behind innovations that could be number two on the list.
Most notable inventions don't cause a quantum leap or immediate big change in the way the human race works. You could say the transistor - but that just made things a bit smaller at the time, though its importance was that - over a relatively short 50 years - it led to the ICs that control just about everything today, and massive changes in lifestyles.
The industrial revolution - mid 1700s to 1800s - also caused some fairly massive lifestyle changes for the "developed" world, though it was a series of less important inventions.
I'd probably get behind Glen with language, that's what really separates us from the rest of the animal kingdom (though I sometimes wonder whether to our advantage or not.... :-) but even that was a series of small developments, not one big step.
A powerful argument can be made for symbolism, the idea that a thing stands for something else, as this is the basis of oral and written language, mathematics, art, ritual, political and social structure.
A separate argument can be made for various tool inventions, whether stone, iron or silicon, as well as an argument for synergy between tools and symbolism. In a way, a tool is a symbol, an item that abstractly takes the place of hand, fingers, teeth, feet or other body part.
A third argument can be made for agriculture, but agriculture would hardly have gotten far without prior development of at least stone tools.
Fuel and energy are only used to power tools.
The computer is a tool whose business it is entirely to store and manipulate symbols. It is now used to design and control most every other kind of tool. The practicality and ubiquity of the computer and its associated communication and control networks are made possible by the silicon integrated circuit. Eventually it might
replace humans altogether, and would certainly be the most disruptive of all innovations if that happens.
But the question was the "greatest" innovation. What one innovation would I be least willing to live without?
There are substitutes available for almost everything except writing. However, writing is of limited use if few can read it, and widespread literacy was not possible until the invention of phonetic writing in 1400 BC. It allows me to know the past, record my thoughts and feelings for the future, forms the basis of symbol and therefore tool manipulation and design, and is my nomination for the "greatest" innovation. It also escapes blame for the perils of war, agriculture, and civilization generally, which far pre-date phonetic writing.
As a mathematician, I am going to vote for nothing, that is, '0', the zero. An innovation of the Middle East (hence the name "Arabic numerals"), it allows both the concept of "none" and works as a placeholder, allowing us to group numbers into sets, whether decimal, octal hexidecimal or binary. Think how much harder arithmetic would be if we still did everything in Roman numerals.
I am going to disagree both with your opinion of the importance of the wheel and your assertion that "the whole of humanity would have habitats on the seas because that is the only place where you move without wheels."
The great civilizations of the western hemisphere (Incas, Mayans, Aztecs, as well as the North American tribes) knew of the wheel but made little use of it; despite the breadth of their society and the importance of communication and transportation, wheeled transport was not used.