In terms of larger projects, the Servo guys always seem to encourage people learning Rust to get involved. One of the projects I'm most interested in is Frank McSherry's timely/differental dataflow repos. I wanted a Rust implementation of Naiad when I first read the paper so I was quite pleased to see one of the authors do an implementation, even if it doesn't seem to be a full time pursuit.
[1] https://github.com/frankmcsherry/timely-dataflow/
Most first-rust-projects that are announced on reddit/discourse seem to be games (Piston), simple web apps, raytracers, or datastructures.