It’s not exactly traditional, but lots of ‘alternative’ style jewelry, like my 90’s style captive bead rings, can be found in titanium or niobium, both of which are used in medical/dental implants.
A dedicated piercing/body art shop can probably point you to a number of options. Piercing jewelry has come a long way in the last 30 years. They can now make seamless, removable rings in a variety of materials, and there’s just generally a lot more available than the old industrial-style stuff like mine.
There’s also the possibility of non-reactive plastics, resins, wood (yes, I’ve seen some beautifully carved all wooden earrings).