There have been several zoochat threats (e.g.,
deep sea creatures in aquariums) on deep sea animals in aquariums and the list of species kept, especially in Japan, is
very long. It is actually impossible to mention an exact limit as it varies depending on animal type. Very roughly, the more complex an animal is the harder it is to bring it to the surface alive. For example: Fish<octopus<sea star. This means that a few aquariums, especially in Japan, now sometimes surpass 300 m for fish and occasionally considerably deeper. There are even commercial enterprices that supply "ordinary" high-end aquarists, at Curaçao and in Indonesia, that approach 250 m too, and may use several days to slowly bring a fish to the surface. In less sensible inverts a few public aquariums and research institutes now can bring up things from around 500 m and are slowly approaching 1 km. Pressure is typically the main problem, but temperature, which often rises quite dramatically as you approach the surface, is an issue too. Compared to these issues light is a secondary problem and can be solved by using red light, a colour most deep water animals can't see (however,
there are exceptions). Another issue is chemistry. In keeping certain deep sea animals alive Monterey Bay realized that oxygen was a problem. Oxygen is necessary for aerobic life, but it also has several toxic properties. This means that true deep-sea animals, which often live in areas with relatively low oxygen, really don't do well in the high oxygen levels of surface water. However, the most extreme is the deep sea vent aquarium at Enoshima. It not only provides warm and cold streams to mimic the seeps, but also adds sodium sulfide that transforms into hydrogen sulfide. Highly toxic to humans and flammable, but necessary for some deep sea vent organisms. Then there's the Abyss Box at Océanopolis Brest, which displayed deep sea shrimp and crabs, and provided a pressure similar to that at 1800 m below the surface. Small in volume at only 16 liters, but a larger can presumably be made larger with sufficient $.
There are only really three aquatic groups that currently are out of reach: Species from the absolutely deepest waters, the largest whales (as mentioned previously) and certain pelagic species (e.g.,
blue shark).