I stand corrected - a filterless tank 'can' be done as long as there is some circulation promoting air infusion and thermal gradient reduction and perhaps increased water changes OR there are plenty of living PLANTS. However, as mentioned, a filter allows a more forgiving environment and maintenance in the removal of organics in suspension and water processing for N2. In an established tank, there is nitrifying bacteria in the substrate, however, it has limited capability relative to water movement through it. I have read that N2 bacteria is an opportunist, dependent on the available ammonia or nitrite being delivered to it, which occurs much better in a filter than the substrate (unless there is a UGF).
So all this gobble-d-goop means that a tank 'can' be maintained without a filter, but for most of us, a filter allows us to maintain a cleaner tank longer without extra effort beyond the weekly water change.
Having said the above, I honestly think that given routine 'normal' maintenance, most of our tanks have much cleaner water than most tropical streams, rivers, ditches, ponds, lakes.... Fish seem to do just fine in very muddy waters (thinking of heavy rains and the amazon river*). So we as hobbyists tend to obsess over water filtration and tank maintenance....we could probably all relax some and our fish would be just fine.
On the other hand, I've seen tanks with very poor water condition and I feel better with crystal clear water.
*Trivia: did you know that the flow of the Amazon river is so great into the ocean that a ship twelve miles out at sea could drop a bucket and retrieve fresh water!? "Believe it or not".