To reach the enderdragon, you will need to be able to kill Endermen; you'll need their enderpearls. However you plan to do that, you'll want the weapons or potions you plan to use against them, that's very individual.
You'll also need to reach the Nether; this means you need obsidian. If you use the frame-building method, you can make do with just some buckets (3 iron per bucket), lava, and water. You then will need to be able to kill blazes; again, it depends on your preferred method. The safest way is probably to build bows and arrows and go at it with a potion of fire resist (once you kill the first one, you can brew these, having collected netherwort and more iron), but you'll might want to sacrifice safety for speed, so you might want to hold off on the potion aspect.
Ender pearl + blaze powder = eye of ender. Finding a stronghold can be done with the eyes, and once you've found one, you can easily get to the end. Killing the ender dragon, of course, is another story, you might want gear for that. Really, it all depends on your fighting style.
So in essence, it depends on your plan for killing the high-level mobs. You can get away with nothing more than iron tools, sure, so long as you're skilled enough with them.