Just some ideas i have had, please tell me if these have been made:

Corner turning or edge turning pyraminx crystal... or both
Skewb that can turn like a 2x2
2 sided 15 puzzle that can flip along edges
An invisible bandaged cube (Has normal stickers, pieces glued together)

Skewb+2x2x2
Possible alternative to the 2-sided 15 puzzle. This idea is different though.
Latch Cube with normal stickers has been suggested (I don't know if someone was that insane though).

I don't get what you mean about the Pyraminx Crystal. That would be a Starminx if you're willing to change the cuts, but otherwise, I don't see how it's possible.

The corner turning Pryaminx Crystal would be a Starminx with that extra edge pieces, i think lol.