What animal can be kept in a tank just under 3 gallons?

In a 3 gallon tank you can't keep much in the way of fish. Some people accept the idea of keeping a single betta in such small space, and while they can survive in extremely limited space, it is not particularly healthy or ideal.

(I remove my recommendation to possibly house tiny schooling fish, because there is not room for a large enough school for their comfort/happiness).

You could add a filter to your tank to keep tiny invertebrates like snails or shrimp (ghost or cherry are pretty good options).

If it's a long (rather than high) tank, you might be able to do a millipede or some kind of insect, but I really think I would personally choose either snails or shrimp if I HAD to keep a pet in such limited space.

Another options is to grow some aquatic plants in the tank. It's really hard to keep any living thing happy and healthy and not-stressed in such limited space, and it's easy for something to go quickly and fatally wrong with water quality or temperature when you're talking about such a small volume of water.

Edit: I had another thought-- some people like to keep marimo in a small tank as a 'pet'. (a moss ball-- it's actually a plant/algae colony, not an animal, but some people consider them lucky and cute, and they require a little more care and special handling than a basic houseplant.) https://en.wikipedia.org/wiki/Marimo

No fish should be in a 3 gallons aquarium. Please at all cost, do not put a betta in there. Neons, guppies and other fishes that have been (suprisingly) suggested in other answers can not live in 3 gallons either. Even 5 gallons is too small for fishes, 10 gallons would be the minimum for a betta to have a decent quality of life. If you put fishes in this they will be sad, lethargic and vulnerable to stress (feeling unsafe) or diseases, and the water quality will be hard to maintain.

Neon tetras for example should be at minimum 6 to be happy since they are social and need schooling to feel safe. They will not school if they're 2 or 3. Bettas are not made to live in a small glass despite what the petshop says. I had also african dwarf frogs before and they should not be in such a small bowl if you want them to live long and happily.

Explanation: Fishes and frogs need to swim, they need to extend their legs (for frogs), to use their full power of swimming and actually use their muscles, reaching an appreciable speed, otherwise they will become weak and ill. Same as the orcas that are kept in captivity and suffer degradation of muscles and lungs because they cannot swim as much as they should. And I don't even talk about how boring it would be for them in such a small place. They are curious and interested, they are not decorations.

What you CAN put: snails or shrimps would be happy in there with a big bunch of java moss or something. It should still be equipped with a filter, or the water should be changed frequently if without.

You can also consider other uses: fill it with dirt for a flowering house-plant. (an aquatic plant could work too). I don't have experience with insects but maybe you could do a cool setup for ants (requires a lid) or land snails. You could keep leeches or worms of some kind also.

Snails like ramshorn or MTS, shrimps like neocaridina, earthworms and etc can all make a sustainable reproducing colony which can be a very interesting project. Such small creatures are better suited for a small bowl.

Consider that a smaller aquarium requires more care and attention to maintain the quality of the water.

It is very hard to get a small tank to be in biological balance and to keep the water quality good, so I suggest you use this tank only for treating your fish with medication.

This can be helpful if you have to treat your fish from a larger tank because using medication in a larger tank will destroy the biological balance and create problems for your fish.

@Manuki does mention using the small tank as a quarantine tank when you get new fish. If you are going to use it for this, you need to cycle the tank before you buy the new fish (you can use water from the larger tank to shorten the time this takes).

A good size for you to begin with is a 60 liters tank, but remember every new tank needs to be properly cycled before you can add any fish. Please take a look here for how this is done.

Remember, even a 60 liters tank will be too small if you want to have more than a handful of small fish of different types. To see the maximum number of fish you can have in a tank, please have a look here; this is the maximum number of fish you can safely have in a well maintained tank.