In the past four months Dropbox has acquired four app companies, effectively turning the filesharing site into an app platform that not only lets people save and share their files, but dig deeply into the content with a host of tools. And yet, those much publicized acquisitions are only the most recent of a string of purchases the company has made over the past year. Swirling around in Dropbox's $10 billion technological DNA are influences from each of these nine acquisitions: Mailbox: Acquired…