Apple - Text Editor on iOS that can edit files from any folder in Dropbox

There's a large number of applications available that can do this.

My preferred text editor happens to be Byword. By default, it'll link to a specific dropbox folder (/Apps/Byword), but you can change this in the app's settings; just setting it to "/" will give access to every folder in your Dropbox.

iA Writer is another popular choice.

If you're looking for something more geared towards code than writing, Textastic is nice.

Additionally, Brett Terpstra has compiled a large list of iOS text editors and features here, though it doesn't specify which apps have full access or if they are limited to a specific App folder in Dropbox.

I've been using 1Writer for some time now and find it to be superb. I have tried all the other popular suggestions.

You can navigate to any Dropbox folder, and you can specify folders to be in your quick access list. Plenty of features like Markdown support and such. Also a powerful actions functionality and customizable keyboard row.

But to speak to your specific need, this was something I too was looking for and found 1Writer to have the right implementation.