One more update. After reading more on the Exchange blogs, I noticed that the problem appears to be triggered by accepting a calendar invite while on the ios device. Indeed, that is what I had done, and I kept getting invites over and over for the same meeting. So my new strategy is to not accept Exchange calendar invites on the ios device.
I had to delete the Exchange account and recreate it to stop the logging looping. This takes only a couple minutes. Be sure to record all your account info first.
I turned off Badge App Icon under notifications->calendar, and set calendar notifications to Banner, so they don't require a response. This way I'm not tempted to accept any invites on the phone. The only reason I accepted the invite anyway was to clear the badge on the icon. This is not a great solution, but may be the easiest work-around until we get an official fix.
I also noticed that I had calendars enabled for other accounts that I never use, like icloud, and yahoo, so I turned those off.
So far, so good. Been running with push enabled for my calendar no problems since yesterday evening. But we'll see how this holds up in a few days.
For those of you who are sure their problems are not Exchange related, I sympathize. Be aware that some email clients, like gmail, may use Exchange in the background. Also, I would suggest you try hooking up to iTunes and doing "restore as new", "restore from backup" if you haven't already. Other than that, I don't have any suggestions, because I am not seeing those other problems. My problem is clearly related to the Exchange calendar issue.