I am looking for an iOS app (iPhone and/or iPad) that lets me store read articles permanently for easy offline browsing.
I have tried a bunch of the ones out there, but somehow they just don't work.
Reeder will do just what you ask for.
In the title you ask for RSS reader offline. But you also request for "offline browsing" and say nothing about RSS in the body.
So, I guess, what you really want is Read It Later (RIL). It doesn't offer offline RSS, but for a good reason: RSS is meant to feed in content, and RIL is content you hand-pick for later offline access.
But if you want specifically RSS, then you should probably pick Byline, as @Weirdo suggested.
I like Byline
It does not store articles forever, however, but it caches up to 250 articles and Web pages they link to for offline reading; very handy for crippled RSS feeds that do not include the whole article, but just a teaser and a link to the Web page.
It syncs with Google Reader and there is a free, ad-supported version and a reasonable priced app without ads.
You can download it from the App Store.
I use Reeder coupled with Instapaper: see something I want to salt away for offline reading, tuck it into Instapaper which caches it nicely.
Instapaper also displays pieces in an "iBooks-like" way for easier reading with typographic control and more.
The nice thing about this combo (one could also use Read Later) is that Reeder lets you set up a simple button to do the salting away on its services panel. Very slick.