I'm sorry you ran into trouble here. Unfortunately, it's hard to say why this keeps happening without some more information. As a first step, you might try uninstalling/reinstalling the Creative Cloud desktop app: Uninstall the Adobe Creative Cloud desktop application
Hopefully that takes care of the issues. If not, I would take a look at our troubleshooting doc for font sync: Troubleshoot font syncing problems in Typekit
There are instructions for sending us diagnostics at the end of that doc if the issue persists. I wouldn't post those here. Send us an email at: firstname.lastname@example.org.
Just want to add that I'm having the same problem, having to go through a long syncing process every time I open Illustrator. I'll check the documentation, but just wanted to pipe in and say this is a problem I experience and hear a lot of other designers complaining about.
I've been having this problem with InDesign CC 2017 for more than one week. When I checked my Typekit account, I was told that all my fonts were still synced, but I was still receiving error messages when I tried to open a document. Out of the blue, I started InDesign late this morning (4/22/17), and everything appears to be resolved: I have all my fonts back. Are anyone else's fonts syncing up again?
Also, I was discussing this problem with someone else, and they suggested that, when I get the fonts to sync again, I might try packaging the document to see if I might still have access to the fonts the next time this happens.
I hope everyone else's problems have resolved now!
I have reinstalled Adobe CC desktop app and still having this problem. It's nearing midnight and i've been fighting with this stupid font issue for about 30 minutes now. Aside from the fact that i'm on a deadline, this new CC platform is crap. I've had to reinstall Adobe CC on every computer i have INDD, PSD, and AI on, all 3 within the a week's time period. Leads me to believe there was a corrupt latest build. Whatever the case, it's extremely frustrating to pay all this money for these cloud-provisioned apps when Adobe can't seem to deploy a stable build. Adobe used to be worth the money. Today, not so much.