Let’s look at the items in the original Zelda.

Armory

• Wooden Sword -> White Sword -> Magical Sword
• Shield -> Magical Shield

You don’t even need a sword, yet if you want to you can keep trading up for better stuff. You can buy a better shield anywhere; the trick is finding the cheapest price. All posturing aside, it is helpful to have some kind of a sword, and you can claim a weak one right in the first screen. It is interesting that the key word is “helpful”, though, not “necessary”. If you want to avoid it, there’s nothing stopping you.

Collectibles

• Heart Containers
• Triforce

There are only so many heart containers, and every time you find one you get an automatic and immediate benefit: another whole heart on the life meter. And the triforce pieces are really just there to check off the dungeons as complete, to give you a reason to delve all the way and beat the boss rather than just rummage around until you find a treasure then skedaddle.

Secondary Inventory

• Raft
• Ladder
• Blue Ring -> Red Ring
• Power Bracelet
• Magic Book
• Magic Key

All “automatic” items, which you don’t need to explicitly use. Only three of them (ladder, raft, power bracelet) really increase the player’s range or abilities, and I don’t think the power bracelet is ever necessary; it’s mostly just useful for opening the warp tunnels. As with the swords and shield, you never need to upgrade your armor, though hey, you’ve got the option. There are enough keys in the game that the magic key is never necessary; it’s just a convenience. And the magic book is an accessory that enhances another item that you don’t even need.

Furthermore, you buy the blue ring in a store and you randomly find the power bracelet in the overworld. You only need to use the raft once. That leaves the ladder as the only really significant upgrade here, and you get it really early on.

Primary Inventory

• Blue Candle-> Red Candle
• Bombs
• Food
• Letter -> Blue Potion -> Red Potion
• Boomerang -> Magical Boomerang
• Bow/Arrow -> Silver Arrow
• Recorder
• Magic Wand

So more important stuff here. You need candles to light the way and burn stuff. Bombs are maybe the most vital item in the game. You need a bow and (some kind of) arrow to beat two bosses. The recorder has its uses. That’s about it, though: candles, bombs, bow and arrow, and recorder. A blue candle, you can buy in almost any shop; the only advantage to the upgrade is that, like the magic key, you can use it indiscriminately. Enemies drop bombs all over, and you can buy them in stores as soon as you’ve the rupees. There are bomb bag upgrades, though once more they’re just an issue of convenience. You find the bow in the first dungeon, and can buy arrows anywhere. I think you need the silver arrow to beat Ganon, but it’s also hidden in Death Mountain, right before the final battle, so.

That only really leaves the recorder as a special and unique item that takes a while to find and is critical to finishing the game. You can buy the food anywhere, and only really need it once. You find the letter for the old women in the overworld, then buy all the potion you want — yet it, like so many other items, is just there for convenience. Enemies drop both boomerangs, and they’re only there to make life easier for the player. The magic wand only exists to be awesome.

The Bare Minimum

So going back through, what do we really need?

• Bombs
• Blue Candle
• Bow/Arrow -> Silver Arrow
• Raft
• Ladder
• Recorder

You can buy half of these in any store. You will have half of them by the time you finish the first dungeon. You find the raft in the third dungeon (which you can enter and complete at any time), and all that really does is grant you access to the fourth dungeon. There you find the ladder, and that opens up almost the entire game. We’ll ignore the silver arrow for the above reasons. That just leaves the recorder; you need it both to beat a boss and to enter another dungeon, and you need to go through a few steps before you can find it. So it’s the only really inaccessible and significant tool in the entire game.

Otherwise, to have all the basic tools, all you need to do is run into level three and grab the raft; use it to enter level four and get the ladder; then buy bombs, a blue candle, and an arrow, and grab the bow from level one. Once you work your way through and find the recorder, you are completely equipped to finish the game.

That’s kind of amazing, really.

As for access:

The entire overworld (save two screens) is accessible from the start. The first three dungeons are immediately and fully accessible. You can enter the fifth and sixth, though without a ladder can’t go far. You enter the eighth by burning a pretty obvious bush, that, should you go exploring, you’ll probably try to burn the first time you see it. If you want, you can beat it right up front — though it’ll be hard.

Burning and bombing are the player’s two main overworld activities. You burn your way into dungeon eight, so you bomb your way (again from a blatantly obvious location) into dungeon nine. You just can’t enter until you have all the triforce pieces, and fair enough.

Four, you need the raft to enter. Which you can claim immediately. Seven, you need the recorder to enter.

So basically, once you have the ladder the only thing aside from the final level that you’re missing out on is duneon seven — again, the recorder.

So again: almost total access within two steps. The only things hindering your progress are danger and a lack of knowledge. There’s practically nothing scripted to hold the player back.

To Put It Numerically:

Collectibles aside and including the armory, there are 26 total inventory items
of those, 8 are required
of those, 5 are freely available from the start
of those, 3 are in shops

of the items that are not freely available:
1 requires only two steps to claim
1 is not necessary until the final battle, and is located near that battle

Again, that only leaves the recorder.

Counting another way:

You start with 1 item
8 items are available in shops
7 items are just lying around to find
10 items are buried in dungeons

Of the items buried in dungeons:
5 are merely upgrades to other items

Of the remaining, distinctive items that are buried in dungeons:
4 are necessary to finish the game

Of those necessary items buried in dungeons:
2 are primary inventory items (that you can use)
2 are secondary (automatic) items
2 (one primary and one secondary) are freely available from the start of the game

Of the remaining necessary items buried in dungeons which are not freely available:
1 can be claimed quickly
1 takes several steps to acquire