Let’s first consider the boundary on the bottom end. The most important piece of advice I can give you in this regard is that it is necessary to do some form of exercise almost every day to optimize your general health. Every man, woman, and child on earth, whether a competitive or recreational runner, whether a runner at all or a non-runner, should aim to exercise every day. The research is very clear on this score. If you exercise daily you will have lower risk of chronic disease, be leaner, and live longer than if you exercise just a few times a week.
This doesn’t mean you have to run every day, however. If you care about running enough to seek some form of progress, you need to run at least three times per week. On the other days you can swim, do yoga, lift weights, whatever. However, if you choose to run only three times per week—and if, again, you care enough about your running to want to improve—you need to make those runs really count. Most weeks those runs should be a tempo run to develop intensive endurance, a speed workout to build speed, and a long run to increase raw endurance. The popular FIRST marathon training program developed at Furman University prescribes a weekly training schedule comprising the three types of runs just mentioned plus two cross-training workouts. In my opinion this system defines the minimum effective training protocol for runners.
The primary reason to run only three times per week is to minimize injury risk. As we all know, running has a high injury rate, and the rate of injury increases with running volume. Many runners cannot run every day without getting injured. If you are such a runner, or if you simply fear getting injured if you run daily, then stick to a schedule of three to four purposeful runs plus a few cross-training workouts per week and feel confident that you are not sacrificing any of the performance you would get from running daily (presuming you actually could run daily without injury).
The most common running frequency for non-elite competitive runners is six to seven times per week (that is, daily with one scheduled day off or daily with rest days taken only as needed). I don’t know of any research addressing the matter, but my experience-based belief is that some runners are better off running daily and not cross-training, others are better off running three or four times a week and cross-training on non-running days, and many runners are able to fare equally well on either schedule. Use factors such as your durability (can you handle daily running?) and your personal preferences (would you rather chew glass than do any form of exercise besides running?) to set your personal routine.