I have a new opening in a pot without drainage where my spider plant resides, which I've decided to move to a more suitable pot for it.
I'd like not to leave the old pot empty, but also not to have to drill holes in it because I don't know if it would break.
Are there any plants that will thrive in a deep pot (about 20cm) that has no drainage?
No, is the short answer. Whilst some plants will grow in waterlogged soil quite happily in open ground, such as in a bog garden, soil in a non draining or waterlogged pot can become hypoxic/anoxic and generally 'sour' because of problems with gas exchange, which means plants will not survive, or at least not do well.
With the type of pot you're describing, they're usually intended to be used as an outer container for an ordinary pot with drainage, so if you want to use it, find a flower pot of the right shape and depth to fit inside, plant into that, and remember to empty the outer pot 30 minutes after watering whatever plant you've used.
Typically a non-draining pot is used for growing succulents (especially cactus.) The idea is to water infrequently (once a month or longer) and completely soak the soil. This mimics the natural environment of many cactus species where rains are infrequent but torrential. You cannot water frequently because the plant will die. Also, The bottom of the pot should be filled with gravel and/or sand to leave room for a little drainage.
Plants are living things and therefore all the cells must breathe. This includes the roots. Even plants which appear to be growing in nothing but water are in fact breathing through the roots, exchanging gases with what is dissolved in the water. As those gases are consumed, the plant eventually has nothing to breathe.
In an aquarium, the plants would soon cease to function if the bubbling air supply was cut off. Plants like watercress can only survive with a constant supply of nutrient rich water that has dissolved gases. Solve the gases problem (with an air pump or mechanical windmill) and you can grow many plants accustomed to being underwater.
