Almost every compiled language has a version of its compiler written in the language.
Is there not an Elm compiler written in Elm?
At the moment the only complete Elm compiler is the "official" one, therefore not named, and it's written in Haskell. I don't know of any implementation of an Elm compiler in Elm.
Over the years that I've read the Elm mailing list(s) I've seen this question come up. Some people expressed interest in having the compiler written in Elm so it can be compiled to JavaScript, and therefore there would be a compiler for Elm in the browser. But so far that's the only compelling argument that's been brought up (IIRC). Which is not enough to make porting the Elm compiler to Elm a priority, the cost-benefit ratio is skewed by the huge effort it would take to port even half of the Haskell libraries that are currently used by elm-compiler.