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.
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.
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 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.