no, I don't think peppers will put out more roots. I think squash will but they put the roots out as they grow since we direct seed.

Interesting! Thank you so much!

Brassicas will.

Peppers will root of you put the in a glass of water for a few weeks. Search "propagating peppers" and you'll find some good examples.

I have no idea if eggplants or tomatillos will propagate or not. Why not try & find out?

The OP isn't propagating from cuttings. S/he wants to know if s/he can plant seedlings deeper when transplanting because they will grow extra roots from the stem where it is buried.

Potatoes do if you happen to plant them from seedlings. Methods like growing potatoes on mounds or in tires can generate more potatoes if you keep covering the stem as it grows. Most stuff like Peppers and Cucerbits would probably rot and dampen off especially if it is too wet

I can say from my own experience that you can plant peppers and eggplants deeper in the soil when you are transplanting out. They won't grow more roots, but the soil will help support the stems if they have gotten leggy. (Don't bury the leaves.) I have never had problems with them rotting. However, I have really really good well drained soil.