I use backtick all the time for Markdown (SE, Github, some blogging software, personal notes, etc.) and code (Ruby shell exec, Bash (although $() is better), Python 2 repr, etc.). I'd say my least used key would be either Insert/Scroll Lock or one of the right-side modifier keys (right Ctrl, right Alt, right Shift).
If I (or my kid, or dog, etc) hit Ctrl and don't realize it, there's nothing to indicate that I did, and it would stay sticky until the next key is pressed. The next key I press may not have the intended effect. If I have to hold down modifier keys, that's not an issue.
Currently this is just an idea. I don't know how to check if an answer is complete, so I'm not sure about the winning criterion.
Make a Pi Game Solver
There is already a question to solve the 24 game, where your program should find a way to get 24 using 4 given integers and addition, subtractio...