How might a shower gamification system look? As mentioned, there need to be sensors. A proximity sensor to see that someone had entered the shower, a flow rate sensor, a temperature sensor and perhaps some kind of way to detect who the showerer is without too much hassle (near field sensor to wrist band or smartphone, voice recognition, image recognition, etc).
There needs to be a (touch) screen and a speaker in order to provide visual and audio feedback to the showerer (whilst also playing music, or displaying the news). As the person gets in, the timer starts, and the combination of time, flow rate and temperature are combined to give an overall score.
Scores are compared against the showerer's personal bests, but also against a family leaderboard, custom leaderboards that the showerer joins (e.g. friends / colleagues), and perhaps local, national or global leaderboards.