No matter how advanced your technology is, your need for and the efficiency of Simple Machines (Wiki) will always be there. Specifically for gears is the ability to sacrifice speed for torque and vise versa (or they can be used in a direct ratio) and in all meshed-gear configurations there is nearly a 100% efficiency in the transfer of energy if we ignore friction.
Two or more simple machines working together to make work easier.
Examples: Wheelbarrow, Can Opener, Bicycle
Two toothed wheels fit together either directly or through a chain or belt so one wheel will turn the other. Some gears may have a screw or a toothed shaft in place of one of the wheels. A gear may also be a combination of toothed wheels that produces a certain speed (such as a bicycle's top gear which makes the bike go fast, and the low gear for slow speed.)
Examples: Clock, Automobile, Drill
I laugh at your x2: (I've no idea what this one was for but huge skycranes need a sleuth gear* that's about that size, as do the spillways on dams, seaway locks, etc.)
.* A sleuth gear is what lets the crane spin in circles (it lays horizontal just underneath the cab). It's like a guide rail and a control surface.
In a facility such as a foundry, the crucible is usually tipped with a hoist but in fully automated factories it's not uncommon to find it suspended in a gear mount. There're just not all that big usually. Metallurgy is a sector that benefits heavily from economy of scale, so I'd have no problem with you up-scaling anything you'd find in a foundry.
If we lived in The Matrix's city of Zion, we'd need some for the dock bay door's counterweights.
Until your society has unlimited electrical power and food replicators, none of your technologies will change, if ever, the basic logistical problems of feeding, clothing, supplying / diverting / removing water, sheltering the population, providing goods of mass production and the tools with which to do so. Or those found in the finished products intrinsically, i.e., a drawbridge:
Now, if you can also control the weather then a lot of these needs disappear, as most of them are required to combat the logistics of delivering goods and providing services. If we're provided all these things and the nanites themselves need nothing (highly implausible) then OK, go gear-less.