Although I've used -with variable success- NN to recognize text patterns (like part number and such), the coolest Neural Net implementation I did was for very simple game which I developed in the context of a challenge/contest for users of Numenta NuPIC framework.
I didn't submit this game for the contest, owing to its incomplete user interface and general "roughness around the edges", the neural network portion of the project, however, was functional and worked rather well.
I realize that Numenta's Hierarchical Temporal Memory (HTM) concept implemented in NuPIC (I was using version 1.3 at the time) is somewhat in departure with traditional Neural Network framworks, but it may be worthy of notice in this SO posting.
The game is one where the player has to learn to communicate with a "pet" (or a "alien being"...) implemented as an HTM network. The mean of communication is by exchanging [imperfect] messages drawn on a small square grid, and to "act" accordingly by pressing a particular action button. The idea is to develop a "language" of sorts to express basic concept (food, water, inside, outside, playing, ball, stick, "I need to sleep" etc.) in a consistent fashion and so that the other party understands them.
The Neural Net portion of the project was derived from the image recognition demo which ships with NuPIC, but included a few twists such as the automatic erasing of the dots that make up the image, a certain amount of time after they are drawn, and also the on-going mix-mode learning/recognition, whereby the demo has these two phases well separated.
The interesting part of this project was how it leveraged the extreme resilience to noise and imprecision in the message being submitted for recognition. HTMs are well known for this feature.
Maybe I should rekindle this, again, very basic / geeky, game and provide it, open-source fashion on Numenta's Site or elsewhere. Another project for when I retire ;-)