The Lightscale Trident is the only other weapon that can be reforged if lost or broken, at Zora's Domain.
You'll have to clear Divine Beast Vah Ruta to get it.
Visit the throne room in Zora's Domain and you'll be told to open the chest for your prize.
Should you lose or break the trident, Dento will refroge it using a Zora Spear, a diamond, and five pieces of flint.
You can get diamonds in Zora's Domain by bringing Ledo - the guy tapping away at a pillar near the shrine - 10 luminous stones. He'll give you two for the first 10, as a side quest reward, and one for every 10 after that.
And that's about as close as you're going to get to unbreakable weapons, so make the most of these small loopholes and enjoy!