"I've trained through illness and injury, run plenty of times when I shouldn't have. I ran on the days my parents passed away and I've run when every one of my four kids was born. I still look forward to running every day, although the trees go by more slowly now."
I'm by no means advocating that you become a streaker of either sort. While I agree that moderate physical activity every day is good for your body and your mind, doing the same type of activity, whether it be running, biking, swimming, or whatever, isn't a good idea for your muscles or joints. Constantly using the same parts of your body in the same way could be your first class ticket to a repetitive stress injury. Not only will that end your streak, but it could put exercise on hold completely. If working out every single day feels good to you, go for it. Just be sure to incorporate different types of exercise that work all the parts of your body. You'll not only prevent injury, but you'll also become stronger overall.