You can bootstrap the compiler. It's a chore but not impossible. More usefully, you can check that your builds are identical to other people's, so at least your compiler isn't uniquely compromised.
I don’t think it’s possible since you’d need the original compilers from the 70’s and bootstrap other compilers up to a modern one. Otherwise your existing compiler could taint your new one.
Many years ago I wrote a C compiler in assembly language. It wasn't hard, and C hasn't changed that much. The complexity in modern compilers is in the optimisation, which you don't need if you're bootstrapping. It's not impossible.
The idea here, is that if you can get a very basic C compiler, you can start building TinyCC, and eventually build a pre-C++ version of GCC, and from there build up to modern GCC. This is a lot easier said than done of course, but not quite as bad as needing the original compilers from the 70s!
It'd be a fun exercise to write a tiny Forth in machine code (sans assembler) and use it to write enough of a C compiler to build tcc, or something along those lines. From there I think you can chain old (but accessible) gcc versions up to modern gcc.
