I think so. I think if you use a straw dipped in bubble solution, it should be able to go inside another bubble without bursting it (if the straw is wet it should be ok), and then you could blow a little bubble inside. Probably wouldn’t work every time, bubbles are tricky like that, but I see no reason why not.
Has anybody (scientists or students!) ever tried this and can report back???
You definitely can, it depends what you mean by a bubble,
Soap bubbles are just the same as a balloon, a think layer of material (soapy water in the case of a bubble) filled with air. You can blow up one balloon inside another and you can get bubbles in bubbles.
a bubble can also be a pocket of gas inside a liquid, or a pocket of another type of liquid inside a liquid. With these types of bubbles you cant make one inside another.