The bigger the tank, the more water it holds, the more water, the more fish, the more fish, the cooler the tank! The more fish, the more fertilizer, the more plants you can grow.
The general consensus for the number of fish a tank can hold is 1 inch of fish per gallon of water. This means that a 10 gallon tank can hold 10 inches of fish, nose to tail. Goldfish are 3-4 inches long, so a 10 gallon aquarium can hold 3 goldfish. This is kinda boring, having so few fish in a tank. So everybody I've ever seen breaks this rule of thumb, and overstocks their tank. But I digress...
I consider any tank or fish bowl under 10 gallons a small aquarium. They can be a lot of trouble, as generally the fish just don't seem to live as long and are prone to massive die-outs when compared to larger tanks. They have to be regularly maintained, as 'letting it go' for a few days on a very small tank like 1 gallon, can have disastrous consequences. They can go from just fine to sick in a matter of hours. The filter systems for these tanks are limited as there isn't much room in the tank for an internal filter, and not much room to hang an over-the-side filter on. External canister filters don't work well here as they are either way to big, or are so weak that they are prone to plug up. Lighting can be challenging, with even the small wattage bulbs causing the water to heat excessively.
Adding a hydroponic component to tanks this small usually means doing something as simple as setting a Peace Plant into the tank. You can also suspend plants over the top of the tank in a Static Solution Culture type system where you take advantage of the aeration bubbles at the top of the tank. If your tank is on the larger size of this size bracket, you may be able to do something like aeroponics or a Nutrient Film Technique if you use a small channel and short, thin hoses. Remember, you don't want to pump much water out of tanks this small as the water level drops quickly. If you top the water off on the tank so it looks good while there is water outside of the tank in they hydroponics system, and the power goes out, a tank overflow is an easy result.
The big benefit to adding hydroponics to a very small system like this is that when used in conjunction with traditional filtering the plants will help regulate the nitrates/nitrites in the aquarium. The levels are likely to run higher then 'normal' but won't spike really high when the filtering is 'used up'. You may be lucky and be able to double or triple the time between cleanings and water changes in the tank - the tank and fish will tell you if you watch closely. One thing to think about here is if you harvest all of whatever your growing in the hydroponics, and re-plant new, the new plants won't be able to absorb as much nutrient as the previous, mature plants - so keep water changes in mind when you replant.
Using ancillary hydroponics fertilizers on a aquaponics system this small may be both required and difficult to grow healthy plants. The fish's fertilizer production may not be enough for the plants at certain stages or at full growth. The instructions on the bottles of fertilizer simply won't cover the small amounts you may need on these very small tanks, maybe needing half a drop or less of fertilizer for your tank! You may need to mix your own diluted version of the fertilizer and use that instead. A bit of math may be involved.
For me, this is in the range of 10 to 55 gallons. These tanks are big enough to have 'real' filter systems. They can go weeks to months without need of cleaning. Tanks of this size are usually set up on stands, have light hoods, and use over-the-back box filters or external canister filters. They are still small enough that they can be moved and set up by a single person.
There are more options to adding a hydroponics system to a tank of this size. They are often times big enough to use more then one hydroponic system type on them at the same time even! If you have a 10 gallon tank, and use a NFT hydroponics system that holds 1 gallon of water, you lower the water level on the tank by about an inch which looks fine. On a 55 gallon tank, you can use an ebb and flow system that has 5 gallons of water capacity and again only affect the water level on the aquarium by an inch or two.
There is a beauty to having 'too many' fish in a tank. By carefully planning and maintaining the hydroponics half of the aquaponics system, you can increase the total water volume of the system possibly by as much as 50% of the base aquarium. Holding water in any trays, trouphs, piping, etc. in such a way that when the power goes out, you don't over-fill your aquarium would be key to safely overstocking your tank.
Aquariums of this size give you a good amount of flexibility in how you set the aquaponics system up. You could simply place a shallow plastic tub right on top of the aquarium and let it drain back into the tank for an ebb and flow system. By using a over-the-back filter designed for deeper tanks, you can hang it right on the side of the tub with the pickup tube able to reach the water in the fish tank. You can use an external canister filter in the place of a regular water pump that is found in hydroponics systems. It pumps and does the mechanical filtration for you at the same time.
The biggest problem with tanks this size is there size. They can get so heavy that many floors can't safely support their weight. If one of these ever does leak, your in for a big mess!
One of the biggest boons of a tank of this size is that many of them use a sump style filter. A sump is basically a small (well, medium sized by my definition above) tank in the stand of the big tank. Water is pumped from the sump into the big tank, where it's designed to overflow back into the sump. When done right, the big tank won't overfill the sump if the pumps quit working due to power outages. Where these filters are awesome for aquaponics in an aquarium, is that the tank always looks full due to any water level fluctuations occurring in the sump. This means that ebb and flow hydroponics systems won't change the level of the aquarium, so it always looks it's best!
Really really big tanks are built with automatic water rotation, auto water level top off, and other such luxuries. These can be leveraged when re-purposed into an aquaponics system because they automatically compensate for the increased evaporation that occurs with the large numbers of plants involved in systems this big. They can also be used with the types of fish that are traditionally used in aquaponics - fish big enough to eat!