What is the maximum amount of energy that can be stored?

The original energy bar will stop automatically refilling energy when it reaches 110 energy. This capacity can be increased to 150 energy if you purchase the extended energy bar for 799 diamonds (offer located by tapping on the "+" sign next to your energy count). However, you can add energy above this capacity, you just won't have the automatic refill energy when you are above 110 or 150. Energy can be added by completing character photo puzzles, activating star boosters, serving and accepting drinks in the lounge, collecting energy from completed landmarks, etc. There is no known maximum for this stored energy.