This is a small nitpick, but I'd say Dropbox competes against sysadmins setting up a corporate network share or FTP server (or even email attachments) for storing documents. They basically automated the sysadmin's job, with a slick interface that was easy to set up and easy to pay for.
Same with Slack. They took the concept of IRC, which many companies use internally, and made it dead simple to set up and use.
In essence, I'd say the competition arising these days is all about UX design. Startups are taking these existing services and making them easier and smoother to use.
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I just improved usability on the site by like 100% for mobile users. In two lines of code. HN, please get on it.