Comment by pcwalton

You can run arbitrary computations on iOS devices if they're written in JavaScript, WebAssembly, or Swift (via Playgrounds). All of these are Turing complete, and all three compile into machine code. What you don't have without an Apple developer account is direct machine code access.

Also note that apps like Pythonista allow you to write programs that call arbitrary Objective-C APIs without permission from Apple. This means that you have a Turing-complete language running unsigned code that can do anything a signed app can do. Your programs do, however, execute slowly.