Neural nets even have their own "universality theorem", meaning any function at all, any sort of process out there in the world, anything, could in theory be approximated by a neural net. Yet everyone just does MNIST :)
I've started trying to get a network to recognise different vowels ("aaahhhhhh", "eeeeee", "ooooooo", etc.). Relatively easy to generate data - you just need your voice and a microphone. Downside is all the NN systems are much more set up for images than sound.
Or what about neural net fingerprint recognition. There must be databases of fingerprints somewhere. Or irises.
Recognise a type of wood from images of its grain?
Or activity recognition from accelerometer data. I think Pebble recently open sourced their recogniser and it was surprisingly not a neural network. I'm sure a neural network could do better. Might be hard to get a decent amount of data here but this could be a good incentive to do exercise!