Yes you can. Automatic water changer... But then, you'd still be doing water changes, it would just require little or no work to get it done.
I think it entirely possible to spend enough money on things that would make water changes fewer and farther between, but eliminate them, not really...
Denitrators work, but cost a fortune and require constant attention. Easier and cheaper to just grab a hose and wait...
Plants above the tank is an interesting idea. I'm planning on trying alot of pothos around my overflows on my new tank. If you live somewhere where water isn't accessible, I bet a well planted tank will be the best bet. Light stocking with tons of plants.....
Well I remember reading one of the large Public aquariums recycling all of their sea water. I think if you had enough things like large carbon blocks and ozone UV and other filtering things I know nothing about you could do it, but the problem is you would have to have such high end and expensive equipment it would not be worth it.
Everything that is in the water has something on the market that can remove it and everything thats not in the water was something on the market to put it in the water. but do you have thousands to spend to recycle the water in a 55 gallon tank?
If you're lazy like I am, understock and use lots and lots of live plants to minimize water changes. All in all though, you have to top off evaporation anyways.
The system they are referring to with terrestrial plants is called aquaponics. There is a lot of great information out there. I have been running a 110 gallon like that with a bunch of big cichlids. You need a lot of plants that grow fast to pull out the nutrients so you don't stress your fish. The other compounds like pheromones that are put into the water by fish can't be removed unless you use activated carbon regularly. The water tends to lose clarity over time and take on a yellowish tinge, but the nitrates will stay at manageable levels. To give you an idea of how many plants, one 300 gallon system should have at least a 4' by 8' gravel bed for growing plants. A lot more effort then some water changes.
I don't think it's possible. Total dissolved solids increases when fish produce waste, when you feed the tank and when plants die. TDS eventually takes a toll on the fish's system. Once they get so high, any new fish will not be able to acclimate to your high TDS environment.
Also, debris softens water and in turn makes the pH drop. This could be combated for a while with crushed coral, aragonite or lacerock to buffer the water, but it doesn't last for ever and you will just end up with even higher TDS as these buffers become less effective.
I think fish such as guppies, or bettas, or ricefish or other short lived fish could live a full life in a large tank without water changes, but even after they die you will need to start the tank over from scratch because the environment will always become uninhabitable over time.