Here are a few examples of robot competitions that are quite entertaining:
The ROBO ONE competition is ‘biped robot entertainment’ (official site). It highlights agility, balance and motion. They are remote controlled but to some extent, in the microcontrol of the movement, autonomous.
12th KondoCup Robot Soccer competition (Source: Robots Dreams). These robots also feature in the ROBO ONE competition.
The Micromouse competition: speed, accuracy, search algorithms, navigation. They get a first run to explore the maze and then ‘speed runs’ to set their best time for the fastest route they found.