There is a list on Wikipedia.
Radar guns use an optical Doppler effect to measure speed. Their acoustic equivalent is used in medicine, where it's called Doppler ultrasound and used to measure blood flow or other sorts of motion in the body.
Animals that use echolocation can use the Doppler shift to gain information about the motion of their surroundings.
A sonic boom occurs when the Doppler shift shifts a frequency to infinity.
I guess one can continue to concoct scenarios. I wonder whether, when you drop a cat off a cliff, you can hear the pitch of its screaming drop as it accelerates. (It's not all that cruel - cats can usually survive a fall at terminal velocity.)
You could use it to determine which way a whale is swimming if you have two boats, both listening to the same whalesong. You could even use the Doppler shift to gain information about the position of the whale because both the whale's position and its velocity contribute to the observed Doppler shift at any given place. (I don't have any information about this actually being done, but it might be an interesting problem to work out the locus of possible whale locations for an observed Doppler shift.)
I was also curious about whether the Doppler effect gives us information on the motion of the crust that moves during an earthquake. I found this reference which suggests it does.