Are there real-life alternatives to the chain+derailleur combo?

Belt drive + Internally Geared Hub is a popular combination.

Existing systems are based around a gates carbon belt drive. See the Breezer Beltway series bikes. Also Reeb Cycles began offering belt drives for mountain bikes when they first came to market.

Also, now there is a breakable belt system called Veer that doesn't require a frame cutout (though uses cogs somewhat different from older belt drive systems) which could be used in configuration with an internally geared hub (when their new pro-version comes out). Currently it appears to be singlespeed only.

Though this is still rather similar to a chain drive system, there is less regular maintenance needed to keep the drivetrain clean as wear is not as damaging on the belt drive system. Performance is different with an IGH hub though.

There are also shaft driven bikes which engage with a standard internally geared hub (instead of a cassette like mechanism). See related SE question with some detail. Any experience with shaft driven bikes?

Pinion drive/gearbox is an enclosed 18-speed drive for the (custom) bottom bracket. No need for a RD in that case.

Belt drive + CVT: the Continuum

An article about the Continuum.

The CVT is a NuVinci N330 which weighs 2.45 kg, quite a bit more than conventional derailleurs, shifters, freehub, and cassette.

I once saw a clever idea of multiple small sprockets between two plates, one of which has radial slots and the other spiral slots, and a mechanism to rotate the two plates relative to one another when a change in gearing is desired. This looks like it might be the patent for it: https://patents.google.com/patent/US3995508A/en

It still uses a chain, but no derallieur, and in theory the small sprockets can be stopped at infinitely many places, making it a CVT.

If you can live with the chain, there is a geared hub by Rohloff (Germany) with 14 gears. Quite expensive, but common on high end foldable bikes.

Yes, fixed gear! The answer may appear facetious but this is what I do for my "utility bike" as it is a very low maintenance setup and completely bomb proof. You soon get used to it and it makes you fit.

If you need gears the relatively mainstream options are either gear hub or derailleur. There are also less common setups with belts or even shafts.

Several solutions have been presented here, but I'll offer my take on this. The answer is no. There is no alternative to the chain/derailleur combo that would offer all of the benefits of a chain/derailleur combo.

Shaft drive, belt drive, internally geared hub, all of them suffer from the same fundamental problem: fixing a flat rear tire is difficult on those.

At least on an internally geared hub, all of them must have some lever against the frame to provide a point of torque, as the hub must be able to convert low torque (and high speed) to high torque (and low speed) or vice versa. The requirement to disconnect and re-connect the torque lever adds to the complexity of fixing a rear tire flat.

Similarly, internally geared hubs must have horizontal dropouts, to be able to adjust chain tension. Adjusting the chain tension on horizontal dropouts is time-consuming. You don't want that to complicate your flat fixing. Also, wrapping the chain around the sprocket is time-consuming. On a derailleur system, you just bend the rear derailleur a little and push the rear wheel in.

Not only that, but what do you do if a chain link breaks? On a derailleur system, you just make the chain a bit shorter using a portable chain tool you carry with you in your emergency toolbox (using a special pin, if using a Shimano chain -- the special pins are so lightweight you can carry several with you).

Also, what if the internally geared hub breaks? Everything on a derailleur is able to be temporarily fixed with just allen keys, chain tool, etc. Even if the chain tensioner breaks, you just select what gear you want and use the bike as a single-speed bike for the rest of the journey. If front or rear derailleur breaks, you use the remaining one that is still functional.

What if one of the only two sprockets (one front, one rear) on an internally geared hub breaks? On a derailleur system, you have at least two sprockets on the front and multiple on the rear. You can use one of the remaining sprockets. On an internally geared bike, you don't have any option.

I don't think any other solution allows the easy temporary workarounds possible on a derailleur system, the easiness of fixing a rear tire flat, the ability to make the chain a link shorter in case a chain link breaks, etc. About the only problem a derailleur system doesn't solve is a freewheel failure. Usually, freewheels give an early warning before completely failing.

The Stringbike is an interesting alternative that supports 19 speeds and claims the wheels are easy to remove. Unlike belt drives, it works in a fundamentally different way than chain+derailleur bikes.

Fallbrook Technologies marketed a CVT (the Nuvinci) for bikes starting around 2006-2007 and came out with a couple of models, getting the weight down to 5.5lbs. I just tried to look them up, and they have since expanded beyond bikes. They spun off the bike CVT stuff to Enviolo (a division of Fallbrook) which is headquartered in Amsterdam even though Fallbrook is an American company (bikes are a bigger deal in Europe, I guess). If you click on the German flag circle, you can get to the english version and download manuals and stuff. I could not see any way to get one in the US and I could not create an account to ask about it. I also didn't see any pricing. The Wikipedia page on Nuvinci has info on ratios (0.5 to 1.9) and such. Found that their FAQ states that US bike shops can order from Fallbrook directly (look under Fallbrooktech.com Contact Us).

It's technically possible to build a delta tricycle in the same way as a three-wheeled car, with a driveshaft and differential, and a gearbox in the drivetrain.

Such an arrangement could be more robust than a typical chain and derailleur, but would also be significantly heavier. An advantage, shared by most delta tricycles, is that working on the rear tyres is easier than on a bicycle.

I'm not aware of any specific examples of this arrangement actually being used. Rather, some early cars were built this way using parts from the established bicycle industry.

Personally, I've never had a chain or sprocket break on me; once, a long-neglected chain was found to have rusted solid, that's all. I have had freewheels latch open, and cables stretch and snap. If breakage often happens to you, you should probably buy stronger parts or re-evaluate your use of them.

Sturmey-Archer have been making hub gears for a very long time! In the UK, most kids' bikes back in the 70s and 80s had 3-speed Sturmey-Archer gears (at least those which had gears anyway). A large number of adult bikes used them too.

Whilst a 3-speed doesn't give you a very good range of gears, the advantages of hub gears has never changed: namely that you can change gears whilst stationary; they need no adjustment; and they are maintenance-free (beyond occasional oiling). For children or for people who don't cycle regularly, these are all very useful. Derailleurs give you better choice of gears so your legs can work more efficiently and speeds can be faster, and parts are cheaper too because they're cheaper to make, but derailleurs simply aren't as user-friendly as hub gears.