I have let my 5D Mark iii go for months without charging batteries without problem, even taking it out and using it occasionally. I have all official Canon gear and I think the longest period yet without charging is probably about 5 months and that was due to use, not loss of charge. If you are losing power in 10 days, you have a short or a defective battery. I would suggest contacting a Canon Service Center for assistance.
Out of camera, I've had a set of LP-E6s sit outside the camera for about 8 months and retain over 90% power. If forget the exact level they were at, but I'm pretty sure it was upwards of 95%. To the best of my knowledge, the off switch means the camera actually turns off completely, so battery life in camera when turned off should be similar and thus is likely measured in years.
One other thought to check, do you have the switch on or off for the controls on the battery grip? I don't believe this matters, but it might be something worth checking. I'm pretty sure they pull power from the camera, but if they pull direct from the batteries, then a button from the controls being pushed while they are switched on could cause a drain.