Broken tooth at gum line, root left behind?

Hello Cheryl and thank you for your question.
The worst that could happen is infection and/or pain.
Often enough pain comes in the late stages when infection spread to the surrounding areas.
I would recommend NOT to wait and the remove the broken root.

Fractured teeth with only their root left in the gingiva can cause a number of problems. Specifically root canal treated teeth can be deceptively painless initially but you must keep in mind that the root left behind is essentially now a remnant of dead tissue that is not even serving a function anymore. 
Usually the gums will slowly grow over the root that is left behind, either partially covering it or completely covering it. When partially covered, there is always a risk of food impaction or rotting of the root remnant leading to a gingival infection and eventually becoming an abscess. When completely covered sometimes the root can be dormant for a while until again some sort of secondary infection sets it.
The take away is, leaving broken roots behind in the gingiva is not a good idea, it can lead to infection, gingival problems, gum disease, and in certain cases odontogenic abscesses. There is also an opportunity for teeth to shift and move into the space where the broken tooth used to be and alter your bite. 
It's highly recommended you have it removed completely to allow the area to heal and then consider some form of replacement, good luck!

You asked what is the worst that can happen? Although quite rare the worst that can happen is that infection can leak down through though the fractured root and eventually spread into your brain and cause death. more common is that the infection will stay more localized but still be a slow drip into the rest of your body. Don't let the absence of pain convince you that there is no cause for concern. Many severe infections and diseases never have a painful component. See a dentist soon to look at your options--good luck