Shouldn't be a problem, as long as you keep an eye on the nutrients.

Prostrate rosemary, chocolate mint - almost any low growing or spreading herb or vegetable should work, really. Just make sure that you don't plant an acid loving tree or bush with an alkaline loving herb, fruit or veggie, and you should be good to go. Other than that, it's just a matter of what you like and how much you like it.

In our garden, we've planted mint, asparagus, herbs and strawberries under our mulched and open/vase shaped fruit trees. So far, they all seem to get along and like it very much.

I read about planting mints under apple trees in a companion planting rag once and both my apple tree and mint did very well last year. I don't know if it is because they were great companions or just because they had good soil, etc.
but I'd check in a companion planting book and see what is supposed to be most beneficial cross referenced to what you like of course.

What about peas and beans? They fix nitrogen and nourish the soil.

Understory plantings will compete with potted trees, but they may still be desirable for a number of reasons, including:

aesthetics
weed suppression
nitrogen fixation
multiple yields, rather than single yield
attract beneficial insects

Mint aggressively commandeers water supply and can quickly take over a pot, so would not be my first choice.

I have purslane growing under moringa and have to cut back the purslane regularly, or the moringa starts to show stress.

I've grown malabar spinach as a vine up small trees with no problems.

Chives, strawberries, clover, oregano would be my first choices.