There is 1 unbreakable weapon in the game, but it has a caveat.
The Sword that Seals the Darkness (also known as the Master Sword) is effectively unbreakable. However, while using it, it will eventually "run out of charge" and can't be used for around 10 minutes, but it will remain in your inventory. The only exception to this is when the blade glows Blue. This means it is near enemies that the game considers to be greatly corrupted. While the blade is glowing blue, it has massively increased durability when striking these enemies. However, it will take normal durability damage when striking rocks/normal enemies/etc, even while glowing blue. It's damage also doubles from 30 to 60.
How to obtain it:
You must go to the large Woods shrouded in fog north of Hyrule Castle. Find your way through them (you cannot paraglider in) until you reach Korok Forest. Sitting in the middle of the forest will be the Master Sword. In order to pull it from the pedestal, you need 13 hearts, as pulling the sword out drains 12 and 3/4 hearts. Temporary hearts do not count for this, you need a full 13 red hearts.
There are a couple of other breakable weapons that are infinitely obtainable.
Each Divine Beast you defeat will reward you with a weapon that represents the Guardian who controls the beast. These can all be rebuilt by NPCs in the city where you obtained them when they break. They all require one generic weapon of the same type, 5 flint, and 1 diamond.
In addition to the below answer, there are now 4 additional spirit orbs (from the second DLC pack) that allow for one further upgrade to either your stamina wheels or your heart containers, bringing to the maximum to 31 heart containers and 3 1/5 stamina wheels.
From various posts like this one, it seems like the maximum number of hearts is 30 and the maximum stamina is 3 full wheels.
There are 120 shrines, the first four spirit orbs being used to obtain the paraglider
That leaves 116 shrines with spirit orbs capable of increasing life/stamina. 116/4 = 29, which means there are a total of 29 upgrades
Correction: these 4 Spirit orbs are not consumed, so they are still able to be used towards upgrades. Thus, 120/4 = 30 possible upgrades.
There are 30 max hearts, and you start with three, so it would take 27 upgrades to max your life
And this thread:
You can upgrade stamina 10 times, each time you upgrade it you get 20% of a wheel. So you can have a maximum of 3 wheels.
Therefore, you can, at most, have 30 hearts and only 2 & 2/5 stamina wheels or 3 full stamina wheels and 27 hearts, as there are 30 upgrades available plus the 4 heart containers given as rewards from the main dungeons.
At this time, it's not certain whether you can fully max out both of these in a single playthrough (the math makes it seem unlikely), but you are able to respec, as explained in this guide:
Make your way to Hateno Village and then head to the bottom of the hill on the other side of the entrance to the village. You should see a sad-looking statue here.
If you talk to this, it’ll start a sidequest that sees the statue steal either a Heart Container or Stamina Vessel from you, but interact with the statue again to reclaim it.
Talk to the statue after this and it’ll offer you a deal whereby you can sell one of your Heart Containers or Stamina Vessels for 100 Rupees. This isn’t as bad as it sounds, though, as if you speak to the statue once more, you’ll be given the option to purchase either a Heart Container or Stamina Vessel for 120 Rupees.
Therefore, you can essentially pay 20 Rupees each time and completely switch your health and stamina upgrades. You can use this statue as many times as you want as long as you have the 20 Rupees necessary.
Best Answer
The maximum number of arrows you can carry is 999 each