Windows only: HP announced earlier this year a specialty line of printer/scanners that will connect to Google Docs. ScanDrop enables the scanner you already have to do the same kind of seamless uploading, too, to Google Docs, Evernote or ScanDrop itself.
The Windows software package works with most modern scanners, along with a wide range of scanners with TWAIN compatibility. Open ScanDrop, load in your papers or images, then do some last-minute light edits with ScanDrop — rotate pages, crop and provide a new folder name, if necessary. Hit the "Upload" button, and your documents are sent on their way.
ScanDrop isn't entirely free, but it offers a 60-day free trial with relatively robust features. Its other relative downside is that documents are uploaded as PDFs to Google Docs, so they can't be searched by content there, though Evernote Premium users can still use that service's OCR powers to get searchable text. ScanDrop's free trial is for Windows systems only.
You could, of course, set up a scanner-app-to-online-storage solution, using some combination of special folders, Dropbox and perhaps other tools. Thought up a solution along those lines, or have one running yourself? Tell us about it in the comments.