We like to say “anything that grows up grows well in aquaponics”, although that does have a few caveats. Acidic pH loving plants like blueberries won’t grow well in an aquaponic system because the fish could not tolerate the level of acidity that those plants require. Also, deep-rooted crops such as corn or wheat likely won’t have enough room to reach maturity in a 1′ deep media bed, so we typically tell folks to leave those large crops in soil outside and use your aquaponic water to fertilize!
Plants in which the edible portions are the leaf (lettuce, kale, chard, mint, and other greens/herbs) will grow faster and with fewer nutrients than fruiting plants. Plants in which we eat the fruit (tomatoes, strawberries, melons, peppers, etc) will take up more space, time, and nutrients to reach maturity. We know of people growing papayas, figs, strawberries, heirloom tomatoes, and cantaloupe, and so much more. We have even successfully grown typically “subterranean” aquaponics plants like beets and radishes, although we prefer to grow these root crops within an external wicking bed. Experiment and let us know what works for you!