How do you export the largest files possible?

Assuming you are using the LRCC desktop app, not the LRCC Web version, when you "export", i.e. use the "Save To" option, and also assuming you are exporting using the Jpeg format, there is a "Size" option where you have 3 choices: Small, Custom, Full Size. Obviously, using the "Full Size" option will do what you want.

One caveat, however....being able to export to full size depends on the image in the cloud being uploaded as a full-size original. That will be the case if you upload to the cloud using any of the LRCC apps, but NOT the case if you upload from LR Classic. In the latter case, only Smart Previews are uploaded, which are 2560 pixels on the long edge.....thus that would be the maximum size available from exporting from LRCC.

All of the above was written on the understanding that when you say you are using "Lightroom CC" I have taken that to mean that you are using the new cloud-based app for which Adobe decided to use the old Lightroom CC name. But if you are in fact using LR CC2015 or LR Classic, then forget all that I wrote and simply look at the "Image Sizing" section in the Export dialog box and make sure "Resize to Fit" is NOT checked....then you'll get full-size exports.