You don't say which version of Service Desk you're on but if it's 7.3.1 you need to apply patch 4592, as detailed Patch for Problem 4592 - Process Designer in 7.3.1. If not, you might just have some duplicate actions, try removing some of the actions, similar to the advice given in Error saving current process: A status instance has multiple transitions leading from it... article.
We're on 7.3.2 ... would I still have to apply the patch or is it "built in"?
I have, this morning, removed everything I had created yesterday (whilst having breakfast) and did an iisreset on the app server.
I copied the process again and when I finally got to the office... started to make the changes to the copied process.
I hit the same error message again.
So starting again, as the process didn't save. I added each new element and after each element was added made sure it had sufficient links, I then saved the process. Then added the next element, linked it in and saved it. Guess what? No errors and I have a process that now looks similar to the one in the test environment!