My referral link, which will give you extra space: https://db.tt/KeG5riq
You get 2Gb free, but additional space can be gained through referrals plus a few activities like Tweeting your love for Dropbox. Dropbox ran online treasure hunts in the past where you could more space for free but unfortunately not in recent years.
My referral link, which will give you extra space: https://copy.com?r=9EliNP
Copy is backed by Barracuda Networks (of hard drive fame) and so they know a thing or two about storage.
Free users get 30 day file revision history and 30 day file recovery, but paid plans are available from $4.99/month with storage beginning at 250Gb. These bump the file recovery and file revision history up to 180 days each.
My referral link, which will give you 0.5Gb extra: https://onedrive.live.com/?invref=c7fe110ca834c114&;invsrc=90
For free you get 5Gb, with referral credit up to an extra 5Gb.
Google do not offer any referral links, but you can sign up at http://google.com/drive
You get 15Gb for free, and this space is shared with your Gmail account as well as other Google apps.
While not a file-based cloud storage service, I truly recommend Google Photos, particularly for your smartphone. Google will let you upload every single photo you have without any storage consideration whatsoever - I kid you not - the only catch being it will resize them if they are above a certain resolution. You can storage your original-sized really high-def photos as part of your Google Drive quota, but no matter how else you might backup your photos you truly will do yourself a favour by checking out Google Photos.
Box does not provide referral credit. You can sign up at http://box.com
Box offers 10Gb for free, albeit with some constraints. You are limited to 10Gb bandwidth per month, and 250Mb maximum file size.