Hard to figure that growing backup issue.. it has hit a number of people.
Mavericks also is very buggy on the Time Machine side. Not at all impressed with the number of issues people are clearly having with it.
How much data do you have total?
So I have couple of bypass the issue suggestions..
1. Are you using ethernet? If not please do so.. you can even remove the TC from the network.. plug it by ethernet Lan port TC, straight into the computer and do the backup in full network isolation.
Over wireless there is too much back and forth for the first backup to run smoothly. If you do it this way actually turn wireless off in the computer.
Make sure ipv6 is set to link local as this is key to working in the new OS.
2. Do a TM backup to a USB drive plugged into the computer. You have USB3 on the latest iMac which is really very fast and it is a good trial to see if there is actually a more basic issue with the backup than the link to the TC.. USB drives of suitable size are cheap and usb3 is pretty well dominate now.. and hardly any more expensive than usb2.
Thank you for your comments. A full back-up on the iMac would be about 650 GB. The partial updates that I have been trying start out at 115MB and keep growing indefinitely. The original back-up never finished nor have any of the follow-up attempts.
I also have a new MacBook Air backing up to the same Time Machine with no problems. I have an older Time Machine (previous version 3TB) that works fine with an older iMac (2008 vintage) and a 2012 MacBook Pro that backs up fine also. All our time machine back-ups are done using WiFi and we have had no problems. Also, the MacBook Air and MacBook Pro are both using Mavericks with no problems on back-ups. The old iMac is still on OS X 10.6.
I have a couple old Seagate external hard drives that I can try your idea about backing up using USB. It might provide a clue.
I'm not sure what you mean by setting "ipv6 to link local"? Can you clarify please.
Thank you again.
When you setup the wireless and ethernet in the computer.. ipv6 will be on automatic.. this is wrong for the latest TC firmware and airport utility. http://support.apple.com/kb/TS4597
Go to the wireless and ethernet setup in preferences and change ipv6 to link local.
Once you do the big backup by ethernet the wireless will function fine for incremental.. it is obviously not estimating the size correctly..
A full back-up on the iMac would be about 650 GB. The partial updates that I have been trying start out at 115MB and keep growing indefinitely. The original back-up never finished nor have any of the follow-up attempts.
The first estimate should be the full backup but it will keep growing as the backup advances.. I would also delete any inprogress files to make sure it is not messing up .. delete even a sparsebundle if there is one for the imac on the TC and start afresh once you have ethernet connection.
See A10 here.
Pondini may have some other clues there to give you an idea of why the backup is slow or not finishing.. there is a lot of issues with TM when it decides not to work.