I thought I would share my experience with our 2013 Mini Cooper Countryman S that would go into limp mode when you accelerated the car.

Basically, when you drove the car and gave it some gas, the speed limiter icon would light up and then the yellow check engine indicator would come on, and then your speed was limited to 20 mph. I did a fair bit of research on the interwebs as to the possible causes, and this was my course of action (in the hope that it might help somebody):

1. Purchased the Bimmerlink app and an OBDLink CX Bimmercode adapter.
2. Plugged in the adapter and scanned the car for errors. 29 errors were found. None were active. Took pictures of the detail of each error for reference.
3. There were codes that indicated problems with various undervoltage conditions, and the battery was a least three years old, so I replaced it (and registered it!).
4. I then wiped all the codes. Confirmed there were zero active codes.
5. Drove the car until limp mode was initiated.
6. Stopped the car and read the codes. All three codes indicated errors with the throttle body.
7. Purchased a Continental OEM throttle body and gasket (and gaskets for the intake manifold).
8. Replaced the throttle body. (What seems like an easy project was actually quite nasty due to a couple of hidden bolts/screws. But I got the job done.)
9. Cleared the codes.
10. Took the car out for an extended test drive. No limp mode, car drives normally.

The Bimmerlink app, while taking a bit of time to get used to, was very useful in diagnosing the conditions of the car. The next time I visit my daughter (she lives 250 miles away) I'll fire up Bimmerlink and see if any other codes have materialized.

P.S. While I was taking out that pesky throttle body, I noticed that the accessory belt is cracked, so I have that replacement in my future.

Hope this helps someone with troubleshooting their vehicle.