There are so many options once Dropbox can theoritcally be accessed through an inbox interface...

Mailbox could be a first step to competing with Messages/gChat. Instead of rolling out their own email, Dropbox can be another type of account Mailbox works with to send pictures, videos, files over data rather than texting or emailing. An odd feature to roll out without acquiring a popular product, but making Dropbox the goto rather than the link attached to something else.