One of the things that the PS4 has over the Xbox One is the ability to easily remove and replace your harddrive. Sony supports this ability and doing so will not void your warranty. That being said, if you are willing to void your Xbox One warranty, the hard drive can be replaced without Microsoft ever touching it.
So get out those screws and open up your Xbox One case! Once you get the plastic shell off you'll see a metal encasing with a Wi-Fi card and speaker on top. Remove these and open up the metal frame and the guts of the Xbox One are there for you to see. You'll notice that the Xbox One's 500GB harddrive is just a standard 2.5 inch SATA II drive. So you can swap this out with any other drive that fits the bill, including solid state drives.
Of course, to do this you will have to propertly format your new drive and that's a little bit more complicated. It actually involves a bit of Linux fenagling and lots of drive formatting. It also requires that you already have removed your Xbox One harddrive and needs more than a bit of terminal skills. If you are up to the challenge, Brian Williams has created a useful tutorial viodeo that walks you through the whole process.
Former Contributing Writer