Evernote lets you store and view PDFs on iPhone / iPod touch (free)
An iPhone / iPod touch application that will let you store and view PDF files, for free, does not exist.
Enter Evernote, a piece of software that purports to be an extension of your brain (it is). Its basic unit of storage is the note, and virtually anything can go inside a note: text, imagery, etc. You can also store a PDF in a note -- and view that note _even when not connected to a network_. Puzzlingly, when you tap a note that contains a PDF, it shows an icon that looks like the PDF needs to be downloaded:
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Tap on that icon, and your PDF will appear. No downloading required. There does appear to be a file size limit, however. I tried to open a 7MB PDF, and Evernote balked. Evernote also limits the amount of storage space that's occupied on its servers, but it's a fairly generous amount and can be upgraded inexpensively.
Check out Evernote!
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I was later told by my friend Ben that the dropbox application also allows you to view files offline, at no cost.