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Buy one of your 5D3 bodies back as a back up camera and second shooting body.
The life of the battery also depends on how often and for how long you use the rear LCD to review images. IS usage was all ready mentioned. Autofocus drive time also uses a fair amount of energy. I agree with others that most of the time you can get many more than the 1120 shots on one battery, but if a shooting session really turns out good, and you are doing a lot of high frame rate bursts, it is not that hard to shoot a lot of images in one day. I have shot over 6000 shots in one day at an air show, on a 1D4, and did use the spare battery that day.
The extra battery is always a good idea, even if you never happen to need it. I always rotate my spare into my camera, for the next time I am out shooting, so that I exercise both of them, and still do also fully charge both of them before I go out to use the camera. The nice dual charger makes that a simple thing to do. Another thing that a spare battery is good for is if you happen to forget to charge your batteries, or end up without enough time to charge them, because you likely have a lot of shots available on your spare.
Enjoy the 1DX. I find it a fantastic camera for a wide variety of shooting.