Using the DMG’s Injuries variant rule (or just having a character who for whatever reason is missing a body part), a character can lose an arm, leg, or eye, and the regenerate spell is the normal way to get it back. But if you don’t have a party member that can cast that, finding someone who can could be quite a hassle. Hence, the question in the title.
Ioun stones are obvious targets for enemies, and a dragon could eat the arm carrying your ring of regeneration. Undying Warlocks get close with their 14th-level feature, but that only lets them reattach body parts, so if their arm does get eaten off, then they’re stuck, too. The UA Armorer subclass for Artificers doesn’t regrow limbs, but it basically gives you an always-on prosthetic limb for any arm or leg that you lose. Eyes (or any other scars or internal injuries), however, aren’t "limbs". I can’t find any other feature (and don’t know of any race) that gets closer to the goal of regeneration without needing to rely on spells or magic items, other than just playing a monster campaign as a troll (AKA not something a "PC" is generally able to do)