Believe it or not, there's an organization that tracks runners who have long running streaks. The United States Running Streak Association defines a running streak as "running at least one continuous mile within each calendar day under one's own body power (without the utilization of any type of health or mechanical aid other than prosthetic devices)". The runs in a streak can occur on roads, tracks, trails, or treadmills. Runners can't use canes, crutches or banisters, or rely on pools or aquatic devices to create artificial buoyancy. When a streaker has run at least a mile every day for a full year, they qualify to be listed on the USRSA's Official USA Active Running Streak List.
If you want to start a running streak, make sure that at least one of your runs every week is very short (you only need to run a mile, after all) and easy to give your body a chance to rest and recover. Also, be careful that you don't push yourself through an injury or illness just to keep your streak alive. You could end up making your injury or illness worse or causing new injuries.
Brag about your running streak in the comments section!