Non-competitive sport for teen

I have an 11 year old son who would be a perfectly happy couch potato if we let him. As a result we have to make him exercise daily, as part of a slate of daily responsibilities he has to complete before free time (which on the weekends includes screen time). He goes to a free afterschool all-sports class at school 2-3 days per week, which isn't super athletic but at least it's movement. On the off days, he has to do 1-3 "Seven Minute Workouts" at home, or 20 minutes on our trampoline. He knows (because we have repeatedly told him) that if he ever wants to stop having exercise as a daily responsibility, he will need to take a regular physical activity that is at least 1 hour/twice per week. We have suggested pre-swim team and karate, because he's also not particularly coordinated or competitive. So far he's going along with the informal home exercise in lieu of the scheduled activity. I'd suggest giving your son the same choice - organized participation a couple times a week, or X at home (could also be jogging twenty minutes with a parent, whatever works for your family), before free time/screen time.

The Berkeley YMCA has a structured program they do for kids wanting PE credit at Berkeley High, but they might do the same for any kid - it's fitness, so not competitive. Martial arts are also good, most aren't competitive, certainly for a while.

You might look into a martial arts program. Both my son and daughter participated in martial arts for many years. Look for a dojo that is low key, non-militant, and comfortable with beginning teens. There are many different styles of martial arts. You might start with aikido. Any good quality dojo will allow a sample free lesson so your son can get a sense of what it's like. My kids (now 17 & 19) grew in confidence, self-esteem, and character as well as in physical strength and coordination. The self-defense aspect tends to be particularly appealing to teens.

Caution: be aware of dojos that are particularly competitive or negative in their discipline techniques.

How about tai kwon do, karate, fencing, yoga,  bicycling- not necessarily competitive but can be done in a group or class. 

Our son did Richmond Sails swim team at the plunge for years (2 x/week).  It is structured but kind and doesn't push the competitive side of swimming.  Their head coach, John, is a junior high school teacher and really gets kids, especially boys.  Our program was: starting at age 8, you must get two days a week of exercise no matter what and until you find something you love, swimming is it.  

Now, at 14, our son bikes and on his own does a 30 min cardio in his room most nights.  We bought him a boxing glove and pad set.  He's got the pads taped to his wall and includes a rigorous set of "boxing" in his workout.  You might consider buying some interesting workout "tools" and having silly fun time with them.  We have a wheel that you stretch to get an ab workout and it has caused much hilarity at our house . . .